Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e'_5
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ✝ φ ψ : SimpleFunc E E
hφ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hψ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
ω : Ω
hωφ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hωψ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
⊢ ∫ (x : Ω), (⇑(φ + ψ) ∘ X 0) x ∂μ = ∫ (a : Ω), (⇑φ ∘ X 0) a + (⇑ψ ∘ X 0) a ∂μ

pp: case e'_5.hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ✝ φ ψ : SimpleFunc E E
hφ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hψ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
ω : Ω
hωφ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hωψ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
⊢ Integrable (⇑φ ∘ X 0) μ

pp: case e'_5.hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ✝ φ ψ : SimpleFunc E E
hφ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hψ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
ω : Ω
hωφ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hωψ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
⊢ Integrable (⇑ψ ∘ X 0) μ

Tactic:
]

Goal state after:
case e'_5
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ✝ φ ψ : SimpleFunc E E
hφ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hψ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
ω : Ω
hωφ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hωψ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
⊢ ∫ (x : Ω), (⇑(φ + ψ) ∘ X 0) x ∂μ = ∫ (a : Ω), (⇑φ ∘ X 0) a + (⇑ψ ∘ X 0) a ∂μ

case e'_5.hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ✝ φ ψ : SimpleFunc E E
hφ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hψ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
ω : Ω
hωφ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hωψ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
⊢ Integrable (⇑φ ∘ X 0) μ

case e'_5.hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ✝ φ ψ : SimpleFunc E E
hφ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hψ : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
ω : Ω
hωφ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))
hωψ : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, ψ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑ψ ∘ X 0) x ∂μ))
⊢ Integrable (⇑ψ ∘ X 0) μ