Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case inl
α : Type u_1
inst✝⁵ : PseudoMetricSpace 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[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (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, ⋃ (_ : 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.192} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 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[>] 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 ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))

pp: case inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace 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[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (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, ⋃ (_ : 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.192} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 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[>] 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 ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))

Tactic:
· specialize hf₀ j
      rw [← hr0, cthickening_zero, (hs (f j)).closure_eq] at hf₀
      exact ⟨d, hf₀, by simp [← hr0]⟩

Goal state after:
case inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace 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[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (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, ⋃ (_ : 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.192} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 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[>] 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 ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))