state
stringlengths 0
159k
| srcUpToTactic
stringlengths 387
167k
| nextTactic
stringlengths 3
9k
| declUpToTactic
stringlengths 22
11.5k
| declId
stringlengths 38
95
| decl
stringlengths 16
1.89k
| file_tag
stringlengths 17
73
|
|---|---|---|---|---|---|---|
case insert
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s✝ : Set α
ι : Type u_5
f : ι → α → F'
i : ι
s : Finset ι
his : i ∉ s
heq : (∀ i ∈ s, Integrable (f i)) → μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m]
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1)
⊢ μ[f i + ∑ x in s, f x|m] =ᵐ[μ] μ[f i|m] + ∑ x in s, μ[f x|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
|
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
|
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.293_0.yd50cWAuCo6hlry
|
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
|
by_cases hm : m ≤ m0
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : ¬m ≤ m0
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
|
swap
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : ¬m ≤ m0
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; ·
|
simp_rw [condexp_of_not_le hm]
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : ¬m ≤ m0
⊢ 0 =ᵐ[μ] c • 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm];
|
simp
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : ¬m ≤ m0
⊢ 0 =ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp;
|
rfl
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp;
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
|
by_cases hμm : SigmaFinite (μ.trim hm)
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
|
swap
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; ·
|
simp_rw [condexp_of_not_sigmaFinite hm hμm]
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ 0 =ᵐ[μ] c • 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm];
|
simp
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ 0 =ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp;
|
rfl
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp;
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
|
haveI : SigmaFinite (μ.trim hm) := hμm
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
|
refine' (condexp_ae_eq_condexpL1 hm _).trans _
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ ↑↑(condexpL1 hm μ (c • f)) =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
|
rw [condexpL1_smul c f]
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ ↑↑(c • condexpL1 hm μ f) =ᵐ[μ] c • μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
|
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ ∀ᵐ (x : α) ∂μ, (μ[f|m]) x = ↑↑(condexpL1 hm μ f) x → ↑↑(c • condexpL1 hm μ f) x = (c • μ[f|m]) x
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
|
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
x : α
hx1 : ↑↑(c • condexpL1 hm μ f) x = (c • ↑↑(condexpL1 hm μ f)) x
hx2 : (μ[f|m]) x = ↑↑(condexpL1 hm μ f) x
⊢ ↑↑(c • condexpL1 hm μ f) x = (c • μ[f|m]) x
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
|
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.303_0.yd50cWAuCo6hlry
|
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
f : α → F'
⊢ μ[-f|m] =ᵐ[μ] -μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
|
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
|
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.316_0.yd50cWAuCo6hlry
|
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
f : α → F'
this : Module ℝ (α → F') := Pi.module α (fun x => F') ℝ
⊢ μ[-f|m] =ᵐ[μ] -μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
|
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
|
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.316_0.yd50cWAuCo6hlry
|
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
f : α → F'
this : Module ℝ (α → F') := Pi.module α (fun x => F') ℝ
⊢ μ[-f|m] = μ[-1 • f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by
|
rw [neg_one_smul ℝ f]
|
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.316_0.yd50cWAuCo6hlry
|
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : Integrable f
hg : Integrable g
⊢ μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
|
simp_rw [sub_eq_add_neg]
|
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.324_0.yd50cWAuCo6hlry
|
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : Integrable f
hg : Integrable g
⊢ μ[f + -g|m] =ᵐ[μ] μ[f|m] + -μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
|
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
|
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.324_0.yd50cWAuCo6hlry
|
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
|
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ : ¬SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
|
swap
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ : ¬SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; ·
|
simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ : ¬SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
⊢ 0 =ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁];
|
rfl
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
|
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
|
by_cases hf : Integrable f μ
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : Integrable f
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : ¬Integrable f
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
|
swap
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : ¬Integrable f
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; ·
|
simp_rw [condexp_undef hf, condexp_zero]
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : ¬Integrable f
⊢ 0 =ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero];
|
rfl
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : Integrable f
⊢ μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
|
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : Integrable f
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, (μ[μ[f|m₂]|m₁]) x ∂μ = ∫ (x : α) in s, (μ[f|m₁]) x ∂μ
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
|
intro s hs _
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s✝ : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, (μ[μ[f|m₂]|m₁]) x ∂μ = ∫ (x : α) in s, (μ[f|m₁]) x ∂μ
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
|
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0✝ : MeasurableSpace α
μ✝ : Measure α
f g : α → F'
s✝ : Set α
m₁ m₂ m0 : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm₂)
hμm₁ this : SigmaFinite (Measure.trim μ (_ : m₁ ≤ m0))
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, (μ[f|m₂]) x ∂μ = ∫ (x : α) in s, (μ[f|m₁]) x ∂μ
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
|
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.330_0.yd50cWAuCo6hlry
|
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
|
by_cases hm : m ≤ m0
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : ¬m ≤ m0
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
|
swap
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : ¬m ≤ m0
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; ·
|
simp_rw [condexp_of_not_le hm]
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : ¬m ≤ m0
⊢ 0 ≤ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm];
|
rfl
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
|
by_cases hμm : SigmaFinite (μ.trim hm)
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
|
swap
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; ·
|
simp_rw [condexp_of_not_sigmaFinite hm hμm]
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ 0 ≤ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm];
|
rfl
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
|
haveI : SigmaFinite (μ.trim hm) := hμm
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
|
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.347_0.yd50cWAuCo6hlry
|
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
⊢ 0 ≤ᵐ[μ] μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
|
by_cases hfint : Integrable f μ
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.359_0.yd50cWAuCo6hlry
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
hfint : Integrable f
⊢ 0 ≤ᵐ[μ] μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
·
|
rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.359_0.yd50cWAuCo6hlry
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
hfint : Integrable f
⊢ μ[0|m] ≤ᵐ[μ] μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
|
exact condexp_mono (integrable_zero _ _ _) hfint hf
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.359_0.yd50cWAuCo6hlry
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
hfint : ¬Integrable f
⊢ 0 ≤ᵐ[μ] μ[f|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
·
|
rw [condexp_undef hfint]
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.359_0.yd50cWAuCo6hlry
|
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
⊢ μ[f|m] ≤ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
|
by_cases hfint : Integrable f μ
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.367_0.yd50cWAuCo6hlry
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
hfint : Integrable f
⊢ μ[f|m] ≤ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
·
|
rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.367_0.yd50cWAuCo6hlry
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
hfint : Integrable f
⊢ μ[f|m] ≤ᵐ[μ] μ[0|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
|
exact condexp_mono hfint (integrable_zero _ _ _) hf
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.367_0.yd50cWAuCo6hlry
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
hfint : ¬Integrable f
⊢ μ[f|m] ≤ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
·
|
rw [condexp_undef hfint]
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.367_0.yd50cWAuCo6hlry
|
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
|
by_cases hm : m ≤ m0
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : ¬m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0;
|
swap
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0;
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : ¬m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; ·
|
simp_rw [condexp_of_not_le hm]
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : ¬m ≤ m0
⊢ 0 =ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm];
|
rfl
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
|
by_cases hμm : SigmaFinite (μ.trim hm)
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm);
|
swap
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm);
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; ·
|
simp_rw [condexp_of_not_sigmaFinite hm hμm]
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ 0 =ᵐ[μ] 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm];
|
rfl
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
|
haveI : SigmaFinite (μ.trim hm) := hμm
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
|
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ ↑↑(condexpL1 hm μ g) =ᵐ[μ] ↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
|
rw [← Lp.ext_iff]
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ condexpL1 hm μ g = condexpL1 hm μ f
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
|
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ ∀ (n : ℕ), condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
|
intro n
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
n : ℕ
⊢ condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
|
ext1
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case h
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
n : ℕ
⊢ ↑↑(condexpL1 hm μ (gs n)) =ᵐ[μ] ↑↑(condexpL1 hm μ (fs n))
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
|
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case h
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
n : ℕ
⊢ μ[fs n|m] =ᵐ[μ] ↑↑(condexpL1 hm μ (fs n))
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
|
exact condexp_ae_eq_condexpL1 hm (fs n)
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
hn_eq : ∀ (n : ℕ), condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n)
⊢ condexpL1 hm μ g = condexpL1 hm μ f
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
|
have hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs
hfs_bound hfs
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
hn_eq : ∀ (n : ℕ), condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n)
hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f))
⊢ condexpL1 hm μ g = condexpL1 hm μ f
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
have hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs
hfs_bound hfs
|
have hcond_gs : Tendsto (fun n => condexpL1 hm μ (gs n)) atTop (𝓝 (condexpL1 hm μ g)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hgs_int n).1) h_int_bound_gs
hgs_bound hgs
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
have hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs
hfs_bound hfs
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
hn_eq : ∀ (n : ℕ), condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n)
hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f))
hcond_gs : Tendsto (fun n => condexpL1 hm μ (gs n)) atTop (𝓝 (condexpL1 hm μ g))
⊢ condexpL1 hm μ g = condexpL1 hm μ f
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
#align measure_theory.condexp_undef MeasureTheory.condexp_undef
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
#align measure_theory.condexp_zero MeasureTheory.condexp_zero
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
· exact hfm
· exact AEStronglyMeasurable'.stronglyMeasurable_mk _
· exact stronglyMeasurable_zero
#align measure_theory.strongly_measurable_condexp MeasureTheory.stronglyMeasurable_condexp
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])
(condexp_ae_eq_condexpL1 hm g).symm)
#align measure_theory.condexp_congr_ae MeasureTheory.condexp_congr_ae
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f := by
refine' ((condexp_congr_ae hf.ae_eq_mk).trans _).trans hf.ae_eq_mk.symm
rw [condexp_of_stronglyMeasurable hm hf.stronglyMeasurable_mk
((integrable_congr hf.ae_eq_mk).mp hfi)]
#align measure_theory.condexp_of_ae_strongly_measurable' MeasureTheory.condexp_of_aestronglyMeasurable'
theorem integrable_condexp : Integrable (μ[f|m]) μ := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm
#align measure_theory.integrable_condexp MeasureTheory.integrable_condexp
/-- The integral of the conditional expectation `μ[f|hm]` over an `m`-measurable set is equal to
the integral of `f` on that set. -/
theorem set_integral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)]
exact set_integral_condexpL1 hf hs
#align measure_theory.set_integral_condexp MeasureTheory.set_integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ := by
suffices ∫ x in Set.univ, (μ[f|m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
simp_rw [integral_univ] at this; exact this
exact set_integral_condexp hm hf (@MeasurableSet.univ _ m)
#align measure_theory.integral_condexp MeasureTheory.integral_condexp
/-- **Uniqueness of the conditional expectation**
If a function is a.e. `m`-measurable, verifies an integrability condition and has same integral
as `f` on all `m`-measurable sets, then it is a.e. equal to `μ[f|hm]`. -/
theorem ae_eq_condexp_of_forall_set_integral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] := by
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => _) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, set_integral_condexp hm hf hs]
#align measure_theory.ae_eq_condexp_of_forall_set_integral_eq MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
by_cases hμ_finite : IsFiniteMeasure μ
swap
· have h : ¬SigmaFinite (μ.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff]
rw [not_isFiniteMeasure_iff] at hμ_finite
rw [condexp_of_not_sigmaFinite bot_le h]
simp only [hμ_finite, ENNReal.top_toReal, inv_zero, zero_smul]
rfl
by_cases hf : Integrable f μ
swap; · rw [integral_undef hf, smul_zero, condexp_undef hf]; rfl
have h_meas : StronglyMeasurable[⊥] (μ[f|⊥]) := stronglyMeasurable_condexp
obtain ⟨c, h_eq⟩ := stronglyMeasurable_bot_iff.mp h_meas
rw [h_eq]
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condexp bot_le hf
simp_rw [h_eq, integral_const] at h_integral
rw [← h_integral, ← smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul]
rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]
exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩
#align measure_theory.condexp_bot' MeasureTheory.condexp_bot'
theorem condexp_bot_ae_eq (f : α → F') :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ := by
rcases eq_zero_or_neZero μ with rfl | hμ
· rw [ae_zero]; exact eventually_bot
· exact eventually_of_forall <| congr_fun (condexp_bot' f)
#align measure_theory.condexp_bot_ae_eq MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine' (condexp_bot' f).trans _; rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]
#align measure_theory.condexp_bot MeasureTheory.condexp_bot
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_add hf hg]
exact (coeFn_add _ _).trans
((condexp_ae_eq_condexpL1 hm _).symm.add (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_add MeasureTheory.condexp_add
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
(hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i in s, f i|m] =ᵐ[μ] ∑ i in s, μ[f i|m] := by
induction' s using Finset.induction_on with i s his heq hf
· rw [Finset.sum_empty, Finset.sum_empty, condexp_zero]
· rw [Finset.sum_insert his, Finset.sum_insert his]
exact (condexp_add (hf i <| Finset.mem_insert_self i s) <|
integrable_finset_sum' _ fun j hmem => hf j <| Finset.mem_insert_of_mem hmem).trans
((EventuallyEq.refl _ _).add (heq fun j hmem => hf j <| Finset.mem_insert_of_mem hmem))
#align measure_theory.condexp_finset_sum MeasureTheory.condexp_finset_sum
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; simp; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; simp; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm _).trans _
rw [condexpL1_smul c f]
refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _
refine' (coeFn_smul c (condexpL1 hm μ f)).mono fun x hx1 hx2 => _
rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2]
#align measure_theory.condexp_smul MeasureTheory.condexp_smul
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] := by
letI : Module ℝ (α → F') := @Pi.module α (fun _ => F') ℝ _ _ fun _ => inferInstance
calc
μ[-f|m] = μ[(-1 : ℝ) • f|m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f|m] := (condexp_smul (-1) f)
_ = -μ[f|m] := neg_one_smul ℝ (μ[f|m])
#align measure_theory.condexp_neg MeasureTheory.condexp_neg
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] := by
simp_rw [sub_eq_add_neg]
exact (condexp_add hf hg.neg).trans (EventuallyEq.rfl.add (condexp_neg g))
#align measure_theory.condexp_sub MeasureTheory.condexp_sub
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] := by
by_cases hμm₁ : SigmaFinite (μ.trim (hm₁₂.trans hm₂))
swap; · simp_rw [condexp_of_not_sigmaFinite (hm₁₂.trans hm₂) hμm₁]; rfl
haveI : SigmaFinite (μ.trim (hm₁₂.trans hm₂)) := hμm₁
by_cases hf : Integrable f μ
swap; · simp_rw [condexp_undef hf, condexp_zero]; rfl
refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' (hm₁₂.trans hm₂)
(fun s _ _ => integrable_condexp.integrableOn)
(fun s _ _ => integrable_condexp.integrableOn) _
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
intro s hs _
rw [set_integral_condexp (hm₁₂.trans hm₂) integrable_condexp hs]
rw [set_integral_condexp (hm₁₂.trans hm₂) hf hs, set_integral_condexp hm₂ hf (hm₁₂ s hs)]
#align measure_theory.condexp_condexp_of_le MeasureTheory.condexp_condexp_of_le
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
#align measure_theory.condexp_mono MeasureTheory.condexp_mono
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono (integrable_zero _ _ _) hfint hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonneg MeasureTheory.condexp_nonneg
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
by_cases hfint : Integrable f μ
· rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
exact condexp_mono hfint (integrable_zero _ _ _) hf
· rw [condexp_undef hfint]
#align measure_theory.condexp_nonpos MeasureTheory.condexp_nonpos
/-- **Lebesgue dominated convergence theorem**: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
`condexpL1`. -/
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
(hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
(hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_setToFun_of_dominated_convergence _ bound_fs hfs_meas h_int_bound_fs hfs_bound hfs
set_option linter.uppercaseLean3 false in
#align measure_theory.tendsto_condexp_L1_of_dominated_convergence MeasureTheory.tendsto_condexpL1_of_dominated_convergence
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
have hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs
hfs_bound hfs
have hcond_gs : Tendsto (fun n => condexpL1 hm μ (gs n)) atTop (𝓝 (condexpL1 hm μ g)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hgs_int n).1) h_int_bound_gs
hgs_bound hgs
|
exact tendsto_nhds_unique_of_eventuallyEq hcond_gs hcond_fs (eventually_of_forall hn_eq)
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m] := by
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm); swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm
rw [← Lp.ext_iff]
have hn_eq : ∀ n, condexpL1 hm μ (gs n) = condexpL1 hm μ (fs n) := by
intro n
ext1
refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)
exact condexp_ae_eq_condexpL1 hm (fs n)
have hcond_fs : Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs
hfs_bound hfs
have hcond_gs : Tendsto (fun n => condexpL1 hm μ (gs n)) atTop (𝓝 (condexpL1 hm μ g)) :=
tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hgs_int n).1) h_int_bound_gs
hgs_bound hgs
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.388_0.yd50cWAuCo6hlry
|
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
(h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
μ[f|m] =ᵐ[μ] μ[g|m]
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
⊢ Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
|
constructor
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mp
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
⊢ Epi f → ∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
·
|
intro _ A a
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
·
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mp
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
a✝ : Epi f
A : C
a : A ⟶ Y
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ a = x ≫ f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
|
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
⊢ (∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f) → Epi f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
·
|
intro hf
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
·
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
hf : ∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f
⊢ Epi f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
|
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr.intro.intro.intro.intro
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
hf : ∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f
A : C
π : A ⟶ Y
hπ : Epi π
a' : A ⟶ X
fac : π ≫ 𝟙 Y = a' ≫ f
⊢ Epi f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
|
rw [comp_id] at fac
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr.intro.intro.intro.intro
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
f : X ⟶ Y
hf : ∀ ⦃A : C⦄ (y : A ⟶ Y), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ f
A : C
π : A ⟶ Y
hπ : Epi π
a' : A ⟶ X
fac : π = a' ≫ f
⊢ Epi f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
|
exact epi_of_epi_fac fac.symm
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
|
Mathlib.CategoryTheory.Abelian.Refinements.81_0.V6xug7mjcHzOwwS
|
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
⊢ Exact S ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
|
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
⊢ (∀ ⦃A : C⦄ (y : A ⟶ cycles S), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ toCycles S) ↔
∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
|
constructor
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
⊢ (∀ ⦃A : C⦄ (y : A ⟶ cycles S), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ toCycles S) →
∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
·
|
intro hS A a ha
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
·
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (y : A ⟶ cycles S), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ toCycles S
A : C
a : A ⟶ S.X₂
ha : a ≫ S.g = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ a = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
|
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mp.intro.intro.intro.intro
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (y : A ⟶ cycles S), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ toCycles S
A : C
a : A ⟶ S.X₂
ha : a ≫ S.g = 0
A' : C
π : A' ⟶ A
hπ : Epi π
x₁ : A' ⟶ S.X₁
fac : π ≫ liftCycles S a ha = x₁ ≫ toCycles S
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ a = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
|
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (y : A ⟶ cycles S), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ toCycles S
A : C
a : A ⟶ S.X₂
ha : a ≫ S.g = 0
A' : C
π : A' ⟶ A
hπ : Epi π
x₁ : A' ⟶ S.X₁
fac : π ≫ liftCycles S a ha = x₁ ≫ toCycles S
⊢ π ≫ a = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by
|
simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
⊢ (∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f) →
∀ ⦃A : C⦄ (y : A ⟶ cycles S), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ toCycles S
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
·
|
intro hS A a
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
·
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
A : C
a : A ⟶ cycles S
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ a = x ≫ toCycles S
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
|
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
A : C
a : A ⟶ cycles S
⊢ (a ≫ iCycles S) ≫ S.g = 0
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by
|
simp
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
case mpr.intro.intro.intro.intro
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
A : C
a : A ⟶ cycles S
A' : C
π : A' ⟶ A
hπ : Epi π
x₁ : A' ⟶ S.X₁
fac : π ≫ a ≫ iCycles S = x₁ ≫ S.f
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ a = x ≫ toCycles S
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
|
exact ⟨A', π, hπ, x₁, by simp only [← cancel_mono S.iCycles, assoc, toCycles_i, fac]⟩
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
A : C
a : A ⟶ cycles S
A' : C
π : A' ⟶ A
hπ : Epi π
x₁ : A' ⟶ S.X₁
fac : π ≫ a ≫ iCycles S = x₁ ≫ S.f
⊢ π ≫ a = x₁ ≫ toCycles S
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
exact ⟨A', π, hπ, x₁, by
|
simp only [← cancel_mono S.iCycles, assoc, toCycles_i, fac]
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
exact ⟨A', π, hπ, x₁, by
|
Mathlib.CategoryTheory.Abelian.Refinements.96_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : Exact S
A : C
x₂ : A ⟶ S.X₂
hx₂ : x₂ ≫ S.g = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
exact ⟨A', π, hπ, x₁, by simp only [← cancel_mono S.iCycles, assoc, toCycles_i, fac]⟩
variable {S}
lemma ShortComplex.Exact.exact_up_to_refinements
(hS : S.Exact) {A : C} (x₂ : A ⟶ S.X₂) (hx₂ : x₂ ≫ S.g = 0) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
|
rw [ShortComplex.exact_iff_exact_up_to_refinements] at hS
|
lemma ShortComplex.Exact.exact_up_to_refinements
(hS : S.Exact) {A : C} (x₂ : A ⟶ S.X₂) (hx₂ : x₂ ≫ S.g = 0) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
|
Mathlib.CategoryTheory.Abelian.Refinements.110_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.Exact.exact_up_to_refinements
(hS : S.Exact) {A : C} (x₂ : A ⟶ S.X₂) (hx₂ : x₂ ≫ S.g = 0) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
S S₁ S₂ : ShortComplex C
hS : ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂), x₂ ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
A : C
x₂ : A ⟶ S.X₂
hx₂ : x₂ ≫ S.g = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S.f
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian groups, we often need to do diagram chases.
Some of these can be carried out in more general abelian categories:
for example, a morphism `X ⟶ Y` in an abelian category `C` is a
monomorphism if and only if for all `A : C`, the induced map
`(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.
Alternatively, the yoneda presheaf functor which sends `X` to the
presheaf of maps `A ⟶ X` for all `A : C` preserves and reflects
monomorphisms.
However, if `p : X ⟶ Y` is an epimorphism in `C` and `A : C`,
`(A ⟶ X) → (A ⟶ Y)` may fail to be surjective (unless `p` is a split
epimorphism).
In this file, the basic result is `epi_iff_surjective_up_to_refinements`
which states that `f : X ⟶ Y` is a morphism in an abelian category,
then it is an epimorphism if and only if for all `y : A ⟶ Y`,
there exists an epimorphism `π : A' ⟶ A` and `x : A' ⟶ X` such
that `π ≫ y = x ≫ f`. In order words, if we allow a precomposition
with an epimorphism, we may lift a morphism to `Y` to a morphism to `X`.
Following unpublished notes by George Bergman, we shall say that the
precomposition by an epimorphism `π ≫ y` is a refinement of `y`. Then,
we get that an epimorphism is a morphism that is "surjective up to refinements".
(This result is similar to the fact that a morphism of sheaves on
a topological space or a site is epi iff sections can be lifted
locally. Then, arguing "up to refinements" is very similar to
arguing locally for a Grothendieck topology (TODO: show that it
corresponds to arguing for the canonical topology on the abelian
category `C` by showing that a morphism in `C` is an epi iff
the corresponding morphisms of sheaves for the canonical
topology is an epi, and that the criteria
`epi_iff_surjective_up_to_refinements` could be deduced from
this equivalence.)
Similarly, it is possible to show that a short complex in an abelian
category is exact if and only if it is exact up to refinements
(see `ShortComplex.exact_iff_exact_up_to_refinements`).
As it is outlined in the documentation of the file
`CategoryTheory.Abelian.Pseudoelements`, the Freyd-Mitchell
embedding theorem implies the existence of a faithful and exact functor `ι`
from an abelian category `C` to the category of abelian groups. If we
define a pseudo-element of `X : C` to be an element in `ι.obj X`, one
may do diagram chases in any abelian category using these pseudo-elements.
However, using this approach would require proving this embedding theorem!
Currently, mathlib contains a weaker notion of pseudo-elements
`CategoryTheory.Abelian.Pseudoelements`. Some theorems can be obtained
using this notion, but there is the issue that for this notion
of pseudo-elements a morphism `X ⟶ Y` in `C` is not determined by
its action on pseudo-elements (see also `Counterexamples/Pseudoelement`).
On the contrary, the approach consisting of working up to refinements
does not require the introduction of other types: we only need to work
with morphisms `A ⟶ X` in `C` which we may consider as being
"sort of elements of `X`". One may carry diagram-chasing by tracking
these morphisms and sometimes introducing an auxiliary epimorphism `A' ⟶ A`.
## References
* George Bergman, A note on abelian categories – translating element-chasing proofs,
and exact embedding in abelian groups (1974)
http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf
-/
namespace CategoryTheory
open Category Limits
variable {C : Type _} [Category C] [Abelian C] {X Y : C} (S : ShortComplex C)
{S₁ S₂ : ShortComplex C}
lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f := by
constructor
· intro _ A a
exact ⟨pullback a f, pullback.fst, inferInstance, pullback.snd, pullback.condition⟩
· intro hf
obtain ⟨A, π, hπ, a', fac⟩ := hf (𝟙 Y)
rw [comp_id] at fac
exact epi_of_epi_fac fac.symm
lemma surjective_up_to_refinements_of_epi (f : X ⟶ Y) [Epi f] {A : C} (y : A ⟶ Y) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f :=
(epi_iff_surjective_up_to_refinements f).1 inferInstance y
lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (S.liftCycles a ha)
exact ⟨A', π, hπ, x₁, by simpa only [assoc, liftCycles_i, toCycles_i] using fac =≫ S.iCycles⟩
· intro hS A a
obtain ⟨A', π, hπ, x₁, fac⟩ := hS (a ≫ S.iCycles) (by simp)
exact ⟨A', π, hπ, x₁, by simp only [← cancel_mono S.iCycles, assoc, toCycles_i, fac]⟩
variable {S}
lemma ShortComplex.Exact.exact_up_to_refinements
(hS : S.Exact) {A : C} (x₂ : A ⟶ S.X₂) (hx₂ : x₂ ≫ S.g = 0) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [ShortComplex.exact_iff_exact_up_to_refinements] at hS
|
exact hS x₂ hx₂
|
lemma ShortComplex.Exact.exact_up_to_refinements
(hS : S.Exact) {A : C} (x₂ : A ⟶ S.X₂) (hx₂ : x₂ ≫ S.g = 0) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [ShortComplex.exact_iff_exact_up_to_refinements] at hS
|
Mathlib.CategoryTheory.Abelian.Refinements.110_0.V6xug7mjcHzOwwS
|
lemma ShortComplex.Exact.exact_up_to_refinements
(hS : S.Exact) {A : C} (x₂ : A ⟶ S.X₂) (hx₂ : x₂ ≫ S.g = 0) :
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f
|
Mathlib_CategoryTheory_Abelian_Refinements
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set E
x y : E
a b : 𝕜
⊢ StrictConvex 𝕜 univ
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
|
intro x _ y _ _ a b _ _ _
|
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
|
Mathlib.Analysis.Convex.Strict.66_0.eLomqYdbrwkwew8
|
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set E
x✝ y✝ : E
a✝⁶ b✝ : 𝕜
x : E
a✝⁵ : x ∈ univ
y : E
a✝⁴ : y ∈ univ
a✝³ : x ≠ y
a b : 𝕜
a✝² : 0 < a
a✝¹ : 0 < b
a✝ : a + b = 1
⊢ a • x + b • y ∈ interior univ
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
|
rw [interior_univ]
|
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
|
Mathlib.Analysis.Convex.Strict.66_0.eLomqYdbrwkwew8
|
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set E
x✝ y✝ : E
a✝⁶ b✝ : 𝕜
x : E
a✝⁵ : x ∈ univ
y : E
a✝⁴ : y ∈ univ
a✝³ : x ≠ y
a b : 𝕜
a✝² : 0 < a
a✝¹ : 0 < b
a✝ : a + b = 1
⊢ a • x + b • y ∈ univ
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
|
exact mem_univ _
|
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
|
Mathlib.Analysis.Convex.Strict.66_0.eLomqYdbrwkwew8
|
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set E
x y : E
a b : 𝕜
t : Set E
hs : StrictConvex 𝕜 s
ht : StrictConvex 𝕜 t
⊢ StrictConvex 𝕜 (s ∩ t)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
#align strict_convex_univ strictConvex_univ
protected nonrec theorem StrictConvex.eq (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y :=
hs.eq hx hy fun H => h <| H ha hb hab
#align strict_convex.eq StrictConvex.eq
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
|
intro x hx y hy hxy a b ha hb hab
|
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
|
Mathlib.Analysis.Convex.Strict.77_0.eLomqYdbrwkwew8
|
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set E
x✝ y✝ : E
a✝ b✝ : 𝕜
t : Set E
hs : StrictConvex 𝕜 s
ht : StrictConvex 𝕜 t
x : E
hx : x ∈ s ∩ t
y : E
hy : y ∈ s ∩ t
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
⊢ a • x + b • y ∈ interior (s ∩ t)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
#align strict_convex_univ strictConvex_univ
protected nonrec theorem StrictConvex.eq (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y :=
hs.eq hx hy fun H => h <| H ha hb hab
#align strict_convex.eq StrictConvex.eq
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
intro x hx y hy hxy a b ha hb hab
|
rw [interior_inter]
|
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
intro x hx y hy hxy a b ha hb hab
|
Mathlib.Analysis.Convex.Strict.77_0.eLomqYdbrwkwew8
|
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set E
x✝ y✝ : E
a✝ b✝ : 𝕜
t : Set E
hs : StrictConvex 𝕜 s
ht : StrictConvex 𝕜 t
x : E
hx : x ∈ s ∩ t
y : E
hy : y ∈ s ∩ t
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
⊢ a • x + b • y ∈ interior s ∩ interior t
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
#align strict_convex_univ strictConvex_univ
protected nonrec theorem StrictConvex.eq (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y :=
hs.eq hx hy fun H => h <| H ha hb hab
#align strict_convex.eq StrictConvex.eq
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
intro x hx y hy hxy a b ha hb hab
rw [interior_inter]
|
exact ⟨hs hx.1 hy.1 hxy ha hb hab, ht hx.2 hy.2 hxy ha hb hab⟩
|
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
intro x hx y hy hxy a b ha hb hab
rw [interior_inter]
|
Mathlib.Analysis.Convex.Strict.77_0.eLomqYdbrwkwew8
|
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s✝ : Set E
x y : E
a b : 𝕜
ι : Sort u_6
s : ι → Set E
hdir : Directed (fun x x_1 => x ⊆ x_1) s
hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)
⊢ StrictConvex 𝕜 (⋃ i, s i)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
#align strict_convex_univ strictConvex_univ
protected nonrec theorem StrictConvex.eq (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y :=
hs.eq hx hy fun H => h <| H ha hb hab
#align strict_convex.eq StrictConvex.eq
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
intro x hx y hy hxy a b ha hb hab
rw [interior_inter]
exact ⟨hs hx.1 hy.1 hxy ha hb hab, ht hx.2 hy.2 hxy ha hb hab⟩
#align strict_convex.inter StrictConvex.inter
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by
|
rintro x hx y hy hxy a b ha hb hab
|
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by
|
Mathlib.Analysis.Convex.Strict.84_0.eLomqYdbrwkwew8
|
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i)
|
Mathlib_Analysis_Convex_Strict
|
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s✝ : Set E
x✝ y✝ : E
a✝ b✝ : 𝕜
ι : Sort u_6
s : ι → Set E
hdir : Directed (fun x x_1 => x ⊆ x_1) s
hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)
x : E
hx : x ∈ ⋃ i, s i
y : E
hy : y ∈ ⋃ i, s i
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
⊢ a • x + b • y ∈ interior (⋃ i, s i)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Order.Group
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
/-!
# Strictly convex sets
This file defines strictly convex sets.
A set is strictly convex if the open segment between any two distinct points lies in its interior.
-/
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Set
open Convex
section OrderedSemiring
variable [OrderedSemiring 𝕜] [TopologicalSpace E] [TopologicalSpace F]
section AddCommMonoid
variable [AddCommMonoid E] [AddCommMonoid F]
section SMul
variable (𝕜)
variable [SMul 𝕜 E] [SMul 𝕜 F] (s : Set E)
/-- A set is strictly convex if the open segment between any two distinct points lies is in its
interior. This basically means "convex and not flat on the boundary". -/
def StrictConvex : Prop :=
s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ interior s
#align strict_convex StrictConvex
variable {𝕜 s}
variable {x y : E} {a b : 𝕜}
theorem strictConvex_iff_openSegment_subset :
StrictConvex 𝕜 s ↔ s.Pairwise fun x y => openSegment 𝕜 x y ⊆ interior s :=
forall₅_congr fun _ _ _ _ _ => (openSegment_subset_iff 𝕜).symm
#align strict_convex_iff_open_segment_subset strictConvex_iff_openSegment_subset
theorem StrictConvex.openSegment_subset (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(h : x ≠ y) : openSegment 𝕜 x y ⊆ interior s :=
strictConvex_iff_openSegment_subset.1 hs hx hy h
#align strict_convex.open_segment_subset StrictConvex.openSegment_subset
theorem strictConvex_empty : StrictConvex 𝕜 (∅ : Set E) :=
pairwise_empty _
#align strict_convex_empty strictConvex_empty
theorem strictConvex_univ : StrictConvex 𝕜 (univ : Set E) := by
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
#align strict_convex_univ strictConvex_univ
protected nonrec theorem StrictConvex.eq (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a • x + b • y ∉ interior s) : x = y :=
hs.eq hx hy fun H => h <| H ha hb hab
#align strict_convex.eq StrictConvex.eq
protected theorem StrictConvex.inter {t : Set E} (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s ∩ t) := by
intro x hx y hy hxy a b ha hb hab
rw [interior_inter]
exact ⟨hs hx.1 hy.1 hxy ha hb hab, ht hx.2 hy.2 hxy ha hb hab⟩
#align strict_convex.inter StrictConvex.inter
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by
rintro x hx y hy hxy a b ha hb hab
|
rw [mem_iUnion] at hx hy
|
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by
rintro x hx y hy hxy a b ha hb hab
|
Mathlib.Analysis.Convex.Strict.84_0.eLomqYdbrwkwew8
|
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i)
|
Mathlib_Analysis_Convex_Strict
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.