mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case inl
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : IsEmpty E
⊢ TendstoInDistribution Y l Z (fun x ↦ μ'') μ'

pp: case inr
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
⊢ TendstoInDistribution Y l Z (fun x ↦ μ'') μ'

Tactic:
·

Goal state after:
case inl
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : IsEmpty E
⊢ TendstoInDistribution Y l Z (fun x ↦ μ'') μ'
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝¹ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_3
inst✝ : Fintype ι
𝓧 : ι → Type u_4
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
f : (i : ι) → 𝓧 i → 𝕜
hX : iIndepFun X μ
mX : ∀ (i : ι), AEMeasurable (X i) μ
hf : ∀ (i : ι), AEStronglyMeasurable (f i) (Measure.map (X i) μ)
⊢ ∫ (x : Ω), (∏ i, f i ∘ X i) x ∂μ = ∏ i, ∫ (x : Ω), (f i ∘ X i) x ∂μ

Tactic:
convert! hX.integral_fun_prod_comp mX hf

Goal state after:
case e'_2
Ω : Type u_1
𝕜 : Type u_2
inst✝¹ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_3
inst✝ : Fintype ι
𝓧 : ι → Type u_4
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
f : (i : ι) → 𝓧 i → 𝕜
hX : iIndepFun X μ
mX : ∀ (i : ι), AEMeasurable (X i) μ
hf : ∀ (i : ι), AEStronglyMeasurable (f i) (Measure.map (X i) μ)
x✝ : Ω
⊢ (∏ i, f i ∘ X i) x✝ = ∏ i, f i (X i x✝)
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case inr.inl
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hF_lip : LipschitzWith 0 F
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))

pp: case inr.inr
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))

Tactic:
·

Goal state after:
case inr.inl
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hF_lip : LipschitzWith 0 F
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_2
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
h : IndepFun (fun a i ↦ AEMeasurable.mk (f ↑i) ⋯ a) (fun a i ↦ AEMeasurable.mk (f ↑i) ⋯ a) κ μ
⊢ ∀ᵐ (a : α) ∂μ, (fun a i ↦ AEMeasurable.mk (f ↑i) ⋯ a) =ᵐ[κ a] fun a i ↦ f (↑i) a

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
h : IndepFun (fun a i ↦ AEMeasurable.mk (f ↑i) ⋯ a) (fun a i ↦ AEMeasurable.mk (f ↑i) ⋯ a) κ μ
⊢ ∀ᵐ (a : α) ∂μ, (fun a i ↦ AEMeasurable.mk (f ↑i) ⋯ a) =ᵐ[κ a] fun a i ↦ f (↑i) a
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case inr.inl
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hF_lip : ∀ (y : E), F x₀ = F y
h_prob : ∀ (n : ι), IsProbabilityMeasure (Measure.map (Y n) μ'')
⊢ Tendsto (fun n ↦ (Measure.map (Y n) μ'').real Set.univ * F x₀) l (𝓝 ((Measure.map Z μ').real Set.univ * F x₀))

Tactic:
have : IsProbabilityMeasure (μ'.map Z) := Measure.isProbabilityMeasure_map hZ

Goal state after:
case inr.inl
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hF_lip : ∀ (y : E), F x₀ = F y
h_prob : ∀ (n : ι), IsProbabilityMeasure (Measure.map (Y n) μ'')
this : IsProbabilityMeasure (Measure.map Z μ')
⊢ Tendsto (fun n ↦ (Measure.map (Y n) μ'').real Set.univ * F x₀) l (𝓝 ((Measure.map Z μ').real Set.univ * F x₀))
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ (Measure.map X μ).prod (Measure.map Y μ) = (Measure.map X' μ').prod (Measure.map Y' μ')

pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X μ

pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y μ

pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X' μ'

pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ (Measure.map X μ).prod (Measure.map Y μ) = (Measure.map X' μ').prod (Measure.map Y' μ')
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X μ

pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y μ

pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X' μ'

pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'

Tactic:
·

Goal state after:
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ (∫ (ω : Ω), id (A.indicator 1 ω) ∂P) • ∫ (ω : Ω), f (X ω) ∂P = P.real A • ∫ (ω : Ω), f (X ω) ∂P

pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ A.indicator 1 ⟂ᵢ[P] X

pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P

Tactic:
·

Goal state after:
Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ (∫ (ω : Ω), id (A.indicator 1 ω) ∂P) • ∫ (ω : Ω), f (X ω) ∂P = P.real A • ∫ (ω : Ω), f (X ω) ∂P
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y μ

pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X' μ'

pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'

Tactic:
·

Goal state after:
case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y μ
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X' μ'

pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'

Tactic:
·

Goal state after:
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X' μ'
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'

Tactic:
·

Goal state after:
case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ A.indicator 1 ⟂ᵢ[P] X

pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P

Tactic:
·

Goal state after:
Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ A.indicator 1 ⟂ᵢ[P] X
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P

Tactic:
·

Goal state after:
Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P
Module: Mathlib.Order.Restriction

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
π : α → Type u_2
inst✝¹ : LocallyFiniteOrderBot α
inst✝ : DecidableEq α
a : α
x : (a : α) → π a
⊢ updateFinset x (Iic a) (frestrictLe a x) = x

Tactic:
simp [frestrictLe]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_1
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h : IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) (AEMeasurable.mk (f k) ⋯) κ μ
⊢ ∀ᵐ (a : α) ∂μ, (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) =ᵐ[κ a] fun a ↦ (f i a, f j a)

pp: case refine_2
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h : IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) (AEMeasurable.mk (f k) ⋯) κ μ
⊢ ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (f k) ⋯ =ᵐ[κ a] f k

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf_Indep : iIndepFun f κ μ
hf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
h : IndepFun (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) (AEMeasurable.mk (f k) ⋯) κ μ
⊢ ∀ᵐ (a : α) ∂μ, (fun a ↦ (AEMeasurable.mk (f i) ⋯ a, AEMeasurable.mk (f j) ⋯ a)) =ᵐ[κ a] fun a ↦ (f i a, f j a)