-- Mathlib/Probability/StrongLaw.lean
theorem strong_law_ae (X : ℕ → Ω → E) (hint : Integrable (X 0) μ)
    (hindep : Pairwise ((IndepFun · · μ) on X))
    (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :
    ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ ↦ (n : ℝ) ⁻¹ • (∑ i ∈ range n, X i ω)) atTop (𝓝 μ[X 0])

Proof state:
case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
this : IsProbabilityMeasure μ
A : ∀ (i : ℕ), Integrable (X i) μ
Y : ℕ → Ω → E := fun i => AEStronglyMeasurable.mk (X i) ⋯
B : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), X n ω = Y n ω
Yint : Integrable (Y 0) μ
C : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
ω : Ω
h₁ : ∀ (n : ℕ), X n ω = Y n ω
h₂ : Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
⊢ Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Tactic:
have : μ[X 0] = μ[Y 0] := integral_congr_ae (AEStronglyMeasurable.ae_eq_mk (A 0).1)

Result:
case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
this✝ : IsProbabilityMeasure μ
A : ∀ (i : ℕ), Integrable (X i) μ
Y : ℕ → Ω → E := fun i => AEStronglyMeasurable.mk (X i) ⋯
B : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), X n ω = Y n ω
Yint : Integrable (Y 0) μ
C : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
ω : Ω
h₁ : ∀ (n : ℕ), X n ω = Y n ω
h₂ : Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
this : ∫ (x : Ω), X 0 x ∂μ = ∫ (x : Ω), Y 0 x ∂μ
⊢ Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Proof state:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Tactic:
by_cases h : ∀ᵐ ω ∂μ, X 0 ω = 0

Result:
case pos
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Proof state:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
this : IsProbabilityMeasure μ
A : ∀ (i : ℕ), Integrable (X i) μ
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Tactic:
let Y : ℕ → Ω → E := fun i ↦ (A i).1.mk (X i)

Result:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
this : IsProbabilityMeasure μ
A : ∀ (i : ℕ), Integrable (X i) μ
Y : ℕ → Ω → E := fun i => AEStronglyMeasurable.mk (X i) ⋯
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))