[GOAL]
α : Type u_1
m0 : MeasurableSpace α
μ μ₁ μ₂ ν ν₁ ν₂ : Measure α
s t : Set α
hs : ↑↑μ s = 0
ht : ↑↑ν t = 0
hst : univ ⊆ s ∪ t
⊢ μ ⟂ₘ ν
[PROOFSTEP]
use toMeasurable μ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs
[GOAL]
case right
α : Type u_1
m0 : MeasurableSpace α
μ μ₁ μ₂ ν ν₁ ν₂ : Measure α
s t : Set α
hs : ↑↑μ s = 0
ht : ↑↑ν t = 0
hst : univ ⊆ s ∪ t
⊢ ↑↑ν (toMeasurable μ s)ᶜ = 0
[PROOFSTEP]
refine' measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx _) ht
[GOAL]
case right
α : Type u_1
m0 : MeasurableSpace α
μ μ₁ μ₂ ν ν₁ ν₂ : Measure α
s t : Set α
hs : ↑↑μ s = 0
ht : ↑↑ν t = 0
hst : univ ⊆ s ∪ t
x : α
hx : x ∈ (toMeasurable μ s)ᶜ
hxs : x ∈ s
⊢ x ∈ toMeasurable μ s
[PROOFSTEP]
exact subset_toMeasurable _ _ hxs
[GOAL]
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
⊢ sum μ ⟂ₘ ν ↔ ∀ (i : ι), μ i ⟂ₘ ν
[PROOFSTEP]
refine' ⟨fun h i => h.mono (le_sum _ _) le_rfl, fun H => _⟩
[GOAL]
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
H : ∀ (i : ι), μ i ⟂ₘ ν
⊢ sum μ ⟂ₘ ν
[PROOFSTEP]
choose s hsm hsμ hsν using H
[GOAL]
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0
hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0
⊢ sum μ ⟂ₘ ν
[PROOFSTEP]
refine' ⟨⋂ i, s i, MeasurableSet.iInter hsm, _, _⟩
[GOAL]
case refine'_1
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0
hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0
⊢ ↑↑(sum μ) (⋂ (i : ι), s i) = 0
[PROOFSTEP]
rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]
[GOAL]
case refine'_1
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0
hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0
⊢ ∀ (i : ι), ↑↑(μ i) (⋂ (b : ι), s b) = 0
[PROOFSTEP]
exact fun i => measure_mono_null (iInter_subset _ _) (hsμ i)
[GOAL]
case refine'_2
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0
hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0
⊢ ↑↑ν (⋂ (i : ι), s i)ᶜ = 0
[PROOFSTEP]
rwa [compl_iInter, measure_iUnion_null_iff]
[GOAL]
α : Type u_1
m0 : MeasurableSpace α
μ μ₁ μ₂ ν ν₁ ν₂ : Measure α
⊢ μ₁ + μ₂ ⟂ₘ ν ↔ μ₁ ⟂ₘ ν ∧ μ₂ ⟂ₘ ν
[PROOFSTEP]
rw [← sum_cond, sum_left, Bool.forall_bool, cond, cond, and_comm]
