[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
set Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
⊢ blimsup Y₁ atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
set Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
⊢ blimsup Y₁ atTop p ≤ᵐ[μ] blimsup Y₂ atTop p
[PROOFSTEP]
let Z : ℕ → Set α := fun i => ⋃ (j) (_ : p j ∧ i ≤ j), Y₂ j
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
⊢ blimsup Y₁ atTop p ≤ᵐ[μ] blimsup Y₂ atTop p
[PROOFSTEP]
suffices ∀ i, μ (atTop.blimsup Y₁ p \ Z i) = 0 by
  rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y₂, iInf_eq_iInter, diff_iInter, measure_iUnion_null_iff]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
this : ∀ (i : ℕ), ↑↑μ (blimsup Y₁ atTop p \ Z i) = 0
⊢ blimsup Y₁ atTop p ≤ᵐ[μ] blimsup Y₂ atTop p
[PROOFSTEP]
rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y₂, iInf_eq_iInter, diff_iInter, measure_iUnion_null_iff]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
⊢ ∀ (i : ℕ), ↑↑μ (blimsup Y₁ atTop p \ Z i) = 0
[PROOFSTEP]
intros i
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
⊢ ↑↑μ (blimsup Y₁ atTop p \ Z i) = 0
[PROOFSTEP]
set W := atTop.blimsup Y₁ p \ Z i
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
⊢ ↑↑μ W = 0
[PROOFSTEP]
by_contra contra
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
⊢ False
[PROOFSTEP]
obtain ⟨d, hd, hd'⟩ :
  ∃ d,
    d ∈ W ∧
      ∀ {ι : Type _} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
        Tendsto δ l (𝓝[>] 0) →
          (∀ᶠ j in l, d ∈ closedBall (w j) (2 * δ j)) →
            Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :=
  Measure.exists_mem_of_measure_ne_zero_of_ae contra (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ W 2)
[GOAL]
case intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd : d ∈ W
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
⊢ False
[PROOFSTEP]
replace hd : d ∈ blimsup Y₁ atTop p := ((mem_diff _).mp hd).1
[GOAL]
case intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
⊢ False
[PROOFSTEP]
obtain ⟨f : ℕ → ℕ, hf⟩ := exists_forall_mem_of_hasBasis_mem_blimsup' atTop_basis hd
[GOAL]
case intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf : ∀ (i : ℕ), d ∈ Y₁ (f i) ∧ p (f i) ∧ f i ∈ Ici i
⊢ False
[PROOFSTEP]
simp only [forall_and] at hf 
[GOAL]
case intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf : (∀ (x : ℕ), d ∈ cthickening (r₁ (f x)) (s (f x))) ∧ (∀ (x : ℕ), p (f x)) ∧ ∀ (x : ℕ), f x ∈ Ici x
⊢ False
[PROOFSTEP]
obtain ⟨hf₀ : ∀ j, d ∈ cthickening (r₁ (f j)) (s (f j)), hf₁, hf₂ : ∀ j, j ≤ f j⟩ := hf
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
⊢ False
[PROOFSTEP]
have hf₃ : Tendsto f atTop atTop :=
  tendsto_atTop_atTop.mpr fun j => ⟨f j, fun i hi => (hf₂ j).trans (hi.trans <| hf₂ i)⟩
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
⊢ False
[PROOFSTEP]
replace hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0) := hr.comp hf₃
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
⊢ False
[PROOFSTEP]
replace hMr : ∀ᶠ j in atTop, M * r₁ (f j) ≤ r₂ (f j) := hf₃.eventually hMr
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
⊢ False
[PROOFSTEP]
replace hf₀ : ∀ j, ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
[GOAL]
case hf₀
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
⊢ ∀ (j : ℕ), ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
intro j
[GOAL]
case hf₀
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
specialize hrp (f j)
[GOAL]
case hf₀
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : OfNat.ofNat 0 (f j) ≤ r₁ (f j)
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
rw [Pi.zero_apply] at hrp 
[GOAL]
case hf₀
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
rcases eq_or_lt_of_le hrp with (hr0 | hrp')
[GOAL]
case hf₀.inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
hr0 : 0 = r₁ (f j)
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
specialize hf₀ j
[GOAL]
case hf₀.inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
hr0 : 0 = r₁ (f j)
hf₀ : d ∈ cthickening (r₁ (f j)) (s (f j))
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
rw [← hr0, cthickening_zero, (hs (f j)).closure_eq] at hf₀ 
[GOAL]
case hf₀.inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
hr0 : 0 = r₁ (f j)
hf₀ : d ∈ s (f j)
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
exact ⟨d, hf₀, by simp [← hr0]⟩
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
hr0 : 0 = r₁ (f j)
hf₀ : d ∈ s (f j)
⊢ d ∈ closedBall d (2 * r₁ (f j))
[PROOFSTEP]
simp [← hr0]
[GOAL]
case hf₀.inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
hrp' : 0 < r₁ (f j)
⊢ ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
[PROOFSTEP]
simpa using
  mem_iUnion₂.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j)) (by positivity) (lt_two_mul_self hrp') (hf₀ j))
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hrp : 0 ≤ r₁ (f j)
hrp' : 0 < r₁ (f j)
⊢ 0 < 2 * r₁ (f j)
[PROOFSTEP]
positivity
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
hf₀ : ∀ (j : ℕ), ∃ w, w ∈ s (f j) ∧ d ∈ closedBall w (2 * r₁ (f j))
⊢ False
[PROOFSTEP]
choose w hw hw' using hf₀
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
⊢ False
[PROOFSTEP]
let C := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
⊢ False
[PROOFSTEP]
have hC : 0 < C := lt_of_lt_of_le zero_lt_one (IsUnifLocDoublingMeasure.one_le_scalingConstantOf μ M⁻¹)
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
⊢ False
[PROOFSTEP]
suffices ∃ η < (1 : ℝ≥0), ∀ᶠ j in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ η
  by
  obtain ⟨η, hη, hη'⟩ := this
  replace hη' : 1 ≤ η := by
    simpa only [ENNReal.one_le_coe_iff] using
      le_of_tendsto (hd' w (fun j => r₁ (f j)) hr <| eventually_of_forall hw') hη'
  exact (lt_self_iff_false _).mp (lt_of_lt_of_le hη hη')
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
this : ∃ η, η < 1 ∧ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑η
⊢ False
[PROOFSTEP]
obtain ⟨η, hη, hη'⟩ := this
[GOAL]
case intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
η : ℝ≥0
hη : η < 1
hη' : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑η
⊢ False
[PROOFSTEP]
replace hη' : 1 ≤ η := by
  simpa only [ENNReal.one_le_coe_iff] using le_of_tendsto (hd' w (fun j => r₁ (f j)) hr <| eventually_of_forall hw') hη'
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.5945} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
η : ℝ≥0
hη : η < 1
hη' : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑η
⊢ 1 ≤ η
[PROOFSTEP]
simpa only [ENNReal.one_le_coe_iff] using le_of_tendsto (hd' w (fun j => r₁ (f j)) hr <| eventually_of_forall hw') hη'
[GOAL]
case intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
η : ℝ≥0
hη : η < 1
hη' : 1 ≤ η
⊢ False
[PROOFSTEP]
exact (lt_self_iff_false _).mp (lt_of_lt_of_le hη hη')
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
⊢ ∃ η, η < 1 ∧ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑η
[PROOFSTEP]
refine' ⟨1 - C⁻¹, tsub_lt_self zero_lt_one (inv_pos.mpr hC), _⟩
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : 0 < C
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
replace hC : C ≠ 0 := ne_of_gt hC
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
let b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
let B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
have h₁ : ∀ j, b j ⊆ B j := fun j => closedBall_subset_closedBall (mul_le_of_le_one_left (hrp (f j)) hM'.le)
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
have h₂ : ∀ j, W ∩ B j ⊆ B j := fun j => inter_subset_right W (B j)
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
have h₃ : ∀ᶠ j in atTop, Disjoint (b j) (W ∩ B j) := by
  apply hMr.mp
  rw [eventually_atTop]
  refine' ⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) _⟩
  change Disjoint (b j) (Z i)ᶜ
  rw [disjoint_compl_right_iff_subset]
  refine' (closedBall_subset_cthickening (hw j) (M * r₁ (f j))).trans ((cthickening_mono hj' _).trans fun a ha => _)
  simp only [mem_iUnion, exists_prop]
  exact ⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
⊢ ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
[PROOFSTEP]
apply hMr.mp
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
⊢ ∀ᶠ (x : ℕ) in atTop, M * r₁ (f x) ≤ r₂ (f x) → Disjoint (b x) (W ∩ B x)
[PROOFSTEP]
rw [eventually_atTop]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
⊢ ∃ a, ∀ (b_1 : ℕ), b_1 ≥ a → M * r₁ (f b_1) ≤ r₂ (f b_1) → Disjoint (b b_1) (W ∩ B b_1)
[PROOFSTEP]
refine' ⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) _⟩
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
⊢ Disjoint (b j) fun a => a ∈ Z i → False
[PROOFSTEP]
change Disjoint (b j) (Z i)ᶜ
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
⊢ Disjoint (b j) (Z i)ᶜ
[PROOFSTEP]
rw [disjoint_compl_right_iff_subset]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
⊢ b j ⊆ Z i
[PROOFSTEP]
refine' (closedBall_subset_cthickening (hw j) (M * r₁ (f j))).trans ((cthickening_mono hj' _).trans fun a ha => _)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
a : α
ha : a ∈ cthickening (r₂ (f j)) (s (f j))
⊢ a ∈ Z i
[PROOFSTEP]
simp only [mem_iUnion, exists_prop]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
a : α
ha : a ∈ cthickening (r₂ (f j)) (s (f j))
⊢ ∃ i_1, (p i_1 ∧ i ≤ i_1) ∧ a ∈ cthickening (r₂ i_1) (s i_1)
[PROOFSTEP]
exact ⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
have h₄ : ∀ᶠ j in atTop, μ (B j) ≤ C * μ (b j) :=
  (hr.eventually (IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul' μ M hM)).mono fun j hj =>
    hj (w j)
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
⊢ ∀ᶠ (j : ℕ) in atTop, ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
refine' (h₃.and h₄).mono fun j hj₀ => _
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
⊢ ↑↑μ (W ∩ closedBall (w j) (r₁ (f j))) / ↑↑μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
change μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
⊢ ↑↑μ (W ∩ B j) / ↑↑μ (B j) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
rcases eq_or_ne (μ (B j)) ∞ with (hB | hB)
[GOAL]
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) = ⊤
⊢ ↑↑μ (W ∩ B j) / ↑↑μ (B j) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
simp [hB]
[GOAL]
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) ≠ ⊤
⊢ ↑↑μ (W ∩ B j) / ↑↑μ (B j) ≤ ↑(1 - C⁻¹)
[PROOFSTEP]
apply ENNReal.div_le_of_le_mul
[GOAL]
case intro.intro.intro.intro.intro.inr.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) ≠ ⊤
⊢ ↑↑μ (W ∩ B j) ≤ ↑(1 - C⁻¹) * ↑↑μ (B j)
[PROOFSTEP]
rw [ENNReal.coe_sub, ENNReal.coe_one, ENNReal.sub_mul fun _ _ => hB, one_mul]
[GOAL]
case intro.intro.intro.intro.intro.inr.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) ≠ ⊤
⊢ ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
[PROOFSTEP]
replace hB : ↑C⁻¹ * μ (B j) ≠ ∞
[GOAL]
case hB
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) ≠ ⊤
⊢ ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
[PROOFSTEP]
refine' ENNReal.mul_ne_top _ hB
[GOAL]
case hB
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) ≠ ⊤
⊢ ↑C⁻¹ ≠ ⊤
[PROOFSTEP]
rwa [ENNReal.coe_inv hC, Ne.def, ENNReal.inv_eq_top, ENNReal.coe_eq_zero]
[GOAL]
case intro.intro.intro.intro.intro.inr.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
⊢ ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
[PROOFSTEP]
obtain ⟨hj₁ : Disjoint (b j) (W ∩ B j), hj₂ : μ (B j) ≤ C * μ (b j)⟩ := hj₀
[GOAL]
case intro.intro.intro.intro.intro.inr.h.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
⊢ ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
[PROOFSTEP]
replace hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)
[GOAL]
case hj₂
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
⊢ ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
[PROOFSTEP]
rw [ENNReal.coe_inv hC, ← ENNReal.div_eq_inv_mul]
[GOAL]
case hj₂
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
⊢ ↑↑μ (B j) / ↑C ≤ ↑↑μ (b j)
[PROOFSTEP]
exact ENNReal.div_le_of_le_mul' hj₂
[GOAL]
case intro.intro.intro.intro.intro.inr.h.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
⊢ ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
[PROOFSTEP]
have hj₃ : ↑C⁻¹ * μ (B j) + μ (W ∩ B j) ≤ μ (B j) :=
  by
  refine' le_trans (add_le_add_right hj₂ _) _
  rw [← measure_union' hj₁ measurableSet_closedBall]
  exact measure_mono (union_subset (h₁ j) (h₂ j))
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
⊢ ↑C⁻¹ * ↑↑μ (B j) + ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j)
[PROOFSTEP]
refine' le_trans (add_le_add_right hj₂ _) _
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
⊢ ↑↑μ (b j) + ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j)
[PROOFSTEP]
rw [← measure_union' hj₁ measurableSet_closedBall]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
⊢ ↑↑μ (b j ∪ W ∩ B j) ≤ ↑↑μ (B j)
[PROOFSTEP]
exact measure_mono (union_subset (h₁ j) (h₂ j))
[GOAL]
case intro.intro.intro.intro.intro.inr.h.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
hj₃ : ↑C⁻¹ * ↑↑μ (B j) + ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j)
⊢ ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
[PROOFSTEP]
replace hj₃ := tsub_le_tsub_right hj₃ (↑C⁻¹ * μ (B j))
[GOAL]
case intro.intro.intro.intro.intro.inr.h.intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ (j : ℕ) (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[Ioi 0] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hB : ↑C⁻¹ * ↑↑μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (b j)
hj₃ : ↑C⁻¹ * ↑↑μ (B j) + ↑↑μ (W ∩ B j) - ↑C⁻¹ * ↑↑μ (B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
⊢ ↑↑μ (W ∩ B j) ≤ ↑↑μ (B j) - ↑C⁻¹ * ↑↑μ (B j)
[PROOFSTEP]
rwa [ENNReal.add_sub_cancel_left hB] at hj₃ 
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
let R₁ i := max 0 (r₁ i)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
let R₂ i := max 0 (r₂ i)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
have hRp : 0 ≤ R₁ := fun i => le_max_left 0 (r₁ i)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
replace hMr : ∀ᶠ i in atTop, M * R₁ i ≤ R₂ i
[GOAL]
case hMr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
⊢ ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
[PROOFSTEP]
refine' hMr.mono fun i hi => _
[GOAL]
case hMr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
i : ℕ
hi : M * r₁ i ≤ r₂ i
⊢ M * R₁ i ≤ R₂ i
[PROOFSTEP]
rw [mul_max_of_nonneg _ _ hM.le, mul_zero]
[GOAL]
case hMr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
i : ℕ
hi : M * r₁ i ≤ r₂ i
⊢ max 0 (M * r₁ i) ≤ R₂ i
[PROOFSTEP]
exact max_le_max (le_refl 0) hi
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
[PROOFSTEP]
simp_rw [← cthickening_max_zero (r₁ _), ← cthickening_max_zero (r₂ _)]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
[PROOFSTEP]
cases' le_or_lt 1 M with hM' hM'
[GOAL]
case inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
[PROOFSTEP]
apply HasSubset.Subset.eventuallyLE
[GOAL]
case inl.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ⊆
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
[PROOFSTEP]
change _ ≤ _
[GOAL]
case inl.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
[PROOFSTEP]
refine' mono_blimsup' (hMr.mono fun i hi _ => cthickening_mono _ (s i))
[GOAL]
case inl.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
i : ℕ
hi : M * R₁ i ≤ R₂ i
x✝ : p i
⊢ max 0 (r₁ i) ≤ max 0 (r₂ i)
[PROOFSTEP]
exact (le_mul_of_one_le_left (hRp i) hM').trans hi
[GOAL]
case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᵐ[μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
[PROOFSTEP]
simp only [← @cthickening_closure _ _ _ (s _)]
[GOAL]
case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (closure (s i))) atTop p ≤ᵐ[μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (closure (s i))) atTop p
[PROOFSTEP]
have hs : ∀ i, IsClosed (closure (s i)) := fun i => isClosed_closure
[GOAL]
case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[Ioi 0] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
hs : ∀ (i : ℕ), IsClosed (closure (s i))
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (closure (s i))) atTop p ≤ᵐ[μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (closure (s i))) atTop p
[PROOFSTEP]
exact blimsup_cthickening_ae_le_of_eventually_mul_le_aux μ p hs (tendsto_nhds_max_right hr) hRp hM hM' hMr
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ} (_ : Tendsto r atTop (𝓝[>] 0)),
    (blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
      (blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) :=
  by
  clear p hr r; intro p r hr
  have hr' : Tendsto (fun i => M * r i) atTop (𝓝[>] 0) := by
    convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero]
  refine' eventuallyLE_antisymm_iff.mpr ⟨_, _⟩
  ·
    exact
      blimsup_cthickening_ae_le_of_eventually_mul_le μ p (inv_pos.mpr hM) hr'
        (eventually_of_forall fun i => by rw [inv_mul_cancel_left₀ hM.ne' (r i)])
  · exact blimsup_cthickening_ae_le_of_eventually_mul_le μ p hM hr (eventually_of_forall fun i => le_refl _)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
clear p hr r
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
intro p r hr
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have hr' : Tendsto (fun i => M * r i) atTop (𝓝[>] 0) := by
  convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
[PROOFSTEP]
convert TendstoNhdsWithinIoi.const_mul hM hr
[GOAL]
case h.e'_5.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ 0 = M * 0
[PROOFSTEP]
simp only [mul_zero]
[GOAL]
case h.e'_5.h.e'_4.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ 0 = M * 0
[PROOFSTEP]
simp only [mul_zero]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
refine' eventuallyLE_antisymm_iff.mpr ⟨_, _⟩
[GOAL]
case refine'_1
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
exact
  blimsup_cthickening_ae_le_of_eventually_mul_le μ p (inv_pos.mpr hM) hr'
    (eventually_of_forall fun i => by rw [inv_mul_cancel_left₀ hM.ne' (r i)])
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
i : ℕ
⊢ M⁻¹ * (M * r i) ≤ r i
[PROOFSTEP]
rw [inv_mul_cancel_left₀ hM.ne' (r i)]
[GOAL]
case refine'_2
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (M * r i) (s i)) atTop p
[PROOFSTEP]
exact blimsup_cthickening_ae_le_of_eventually_mul_le μ p hM hr (eventually_of_forall fun i => le_refl _)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
let r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / ((i : ℝ) + 1)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have hr' : Tendsto r' atTop (𝓝[>] 0) :=
  by
  refine'
    tendsto_nhdsWithin_iff.mpr ⟨Tendsto.if' hr tendsto_one_div_add_atTop_nhds_0_nat, eventually_of_forall fun i => _⟩
  by_cases hi : 0 < r i
  · simp [hi]
  · simp only [hi, one_div, mem_Ioi, if_false, inv_pos]; positivity
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
⊢ Tendsto r' atTop (𝓝[Ioi 0] 0)
[PROOFSTEP]
refine'
  tendsto_nhdsWithin_iff.mpr ⟨Tendsto.if' hr tendsto_one_div_add_atTop_nhds_0_nat, eventually_of_forall fun i => _⟩
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
⊢ r' i ∈ Ioi 0
[PROOFSTEP]
by_cases hi : 0 < r i
[GOAL]
case pos
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : 0 < r i
⊢ r' i ∈ Ioi 0
[PROOFSTEP]
simp [hi]
[GOAL]
case neg
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ r' i ∈ Ioi 0
[PROOFSTEP]
simp only [hi, one_div, mem_Ioi, if_false, inv_pos]
[GOAL]
case neg
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ 0 < ↑i + 1
[PROOFSTEP]
positivity
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₀ : ∀ i, p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i) := by rintro i ⟨-, hi⟩; congr! 1;
  change r i = ite (0 < r i) (r i) _; simp [hi]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
⊢ ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
[PROOFSTEP]
rintro i ⟨-, hi⟩
[GOAL]
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ cthickening (r i) (s i) = cthickening (r' i) (s i)
[PROOFSTEP]
congr! 1
[GOAL]
case intro.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ r i = r' i
[PROOFSTEP]
change r i = ite (0 < r i) (r i) _
[GOAL]
case intro.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ r i = if 0 < r i then r i else 1 / (↑i + 1)
[PROOFSTEP]
simp [hi]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₁ : ∀ i, p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) := by rintro i ⟨-, hi⟩;
  simp only [hi, mul_ite, if_true]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
⊢ ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
[PROOFSTEP]
rintro i ⟨-, hi⟩
[GOAL]
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
i : ℕ
hi : 0 < r i
⊢ cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
[PROOFSTEP]
simp only [hi, mul_ite, if_true]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₂ : ∀ i, p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i) :=
  by
  rintro i ⟨-, hi⟩
  have hi' : M * r i ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi
  rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
⊢ ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
[PROOFSTEP]
rintro i ⟨-, hi⟩
[GOAL]
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)
[PROOFSTEP]
have hi' : M * r i ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi
[GOAL]
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
hi' : M * r i ≤ 0
⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)
[PROOFSTEP]
rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
have hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0 := by ext i; simp [← and_or_left, lt_or_le 0 (r i)]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
⊢ p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
[PROOFSTEP]
ext i
[GOAL]
case h.a
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
i : ℕ
⊢ p i ↔ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
[PROOFSTEP]
simp [← and_or_left, lt_or_le 0 (r i)]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
rw [hp, blimsup_or_eq_sup, blimsup_or_eq_sup]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
[PROOFSTEP]
simp only [sup_eq_union]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
[PROOFSTEP]
rw [blimsup_congr (eventually_of_forall h₀), blimsup_congr (eventually_of_forall h₁),
  blimsup_congr (eventually_of_forall h₂)]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪
      blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x ∧ r x ≤ 0) =ᵐ[μ]
    (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
[PROOFSTEP]
exact ae_eq_set_union (this (fun i => p i ∧ 0 < r i) hr') (ae_eq_refl _)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
refine' eventuallyLE_antisymm_iff.mpr ⟨_, HasSubset.Subset.eventuallyLE (_ : _ ≤ _)⟩
[GOAL]
case refine'_1
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq μ p s (@one_half_pos ℝ _) r hr).symm EventuallyEq.rfl]
[GOAL]
case refine'_1
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
apply HasSubset.Subset.eventuallyLE
[GOAL]
case refine'_1.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p ⊆ blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
change _ ≤ _
[GOAL]
case refine'_1.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p ≤ blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
refine' mono_blimsup' (hr'.mono fun i hi pi => cthickening_subset_thickening' (hi pi) _ (s i))
[GOAL]
case refine'_1.h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
i : ℕ
hi : p i → 0 < r i
pi : p i
⊢ 1 / 2 * r i < r i
[PROOFSTEP]
nlinarith [hi pi]
[GOAL]
case refine'_2
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => thickening (r i) (s i)) atTop p ≤ blimsup (fun i => cthickening (r i) (s i)) atTop p
[PROOFSTEP]
exact mono_blimsup fun i _ => thickening_subset_cthickening _ _
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₁ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr'
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₂ := blimsup_cthickening_mul_ae_eq μ p s hM r hr
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
replace hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
[GOAL]
case hr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ Tendsto (fun i => M * r i) atTop (𝓝 0)
[PROOFSTEP]
convert hr.const_mul M
[GOAL]
case h.e'_5.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ 0 = M * 0
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
replace hr' : ∀ᶠ i in atTop, p i → 0 < M * r i := hr'.mono fun i hi hip => mul_pos hM (hi hip)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₃ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr'
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
h₁ :
  (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
h₃ :
  (blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i) =ᵐ[μ]
    blimsup (fun i => thickening (M * r i) (s i)) atTop fun i => p i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
exact h₃.symm.trans (h₂.trans h₁)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
let q : ℕ → Prop := fun i => p i ∧ 0 < r i
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q :=
  by
  refine' blimsup_congr' (eventually_of_forall fun i h => _)
  replace hi : 0 < r i; · contrapose! h; apply thickening_of_nonpos h
  simp only [hi, iff_self_and, imp_true_iff]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
⊢ blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
[PROOFSTEP]
refine' blimsup_congr' (eventually_of_forall fun i h => _)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
i : ℕ
h : thickening (r i) (s i) ≠ ⊥
⊢ p i ↔ q i
[PROOFSTEP]
replace hi : 0 < r i
[GOAL]
case hi
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
i : ℕ
h : thickening (r i) (s i) ≠ ⊥
⊢ 0 < r i
[PROOFSTEP]
contrapose! h
[GOAL]
case hi
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
i : ℕ
h : r i ≤ 0
⊢ thickening (r i) (s i) = ⊥
[PROOFSTEP]
apply thickening_of_nonpos h
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
i : ℕ
h : thickening (r i) (s i) ≠ ⊥
hi : 0 < r i
⊢ p i ↔ q i
[PROOFSTEP]
simp only [hi, iff_self_and, imp_true_iff]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
have h₂ :
  blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q :=
  by
  refine' blimsup_congr' (eventually_of_forall fun i h => _)
  replace h : 0 < r i; · rw [← zero_lt_mul_left hM]; contrapose! h; apply thickening_of_nonpos h
  simp only [h, iff_self_and, imp_true_iff]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q
[PROOFSTEP]
refine' blimsup_congr' (eventually_of_forall fun i h => _)
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
i : ℕ
h : thickening (M * r i) (s i) ≠ ⊥
⊢ p i ↔ q i
[PROOFSTEP]
replace h : 0 < r i
[GOAL]
case h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
i : ℕ
h : thickening (M * r i) (s i) ≠ ⊥
⊢ 0 < r i
[PROOFSTEP]
rw [← zero_lt_mul_left hM]
[GOAL]
case h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
i : ℕ
h : thickening (M * r i) (s i) ≠ ⊥
⊢ 0 < M * r i
[PROOFSTEP]
contrapose! h
[GOAL]
case h
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
i : ℕ
h : M * r i ≤ 0
⊢ thickening (M * r i) (s i) = ⊥
[PROOFSTEP]
apply thickening_of_nonpos h
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
i : ℕ
h : 0 < r i
⊢ p i ↔ q i
[PROOFSTEP]
simp only [h, iff_self_and, imp_true_iff]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
h₂ : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p
[PROOFSTEP]
rw [h₁, h₂]
[GOAL]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun i => p i ∧ 0 < r i
h₁ : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q
h₂ : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop q =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop q
[PROOFSTEP]
exact blimsup_thickening_mul_ae_eq_aux μ q s hM r hr (eventually_of_forall fun i hi => hi.2)
