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

Goal state before:
pp: case e'_3
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝⁴ : TopologicalSpace E
F : Type u_6
inst✝³ : OpensMeasurableSpace E
inst✝² : TopologicalSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
g : E → F
hg : Continuous[inst✝⁴, inst✝²] g
h : TendstoInDistribution X l Z μ μ'
x✝ : ι
⊢ ⟨Measure.map (g ∘ X x✝) (μ x✝), ⋯⟩ = ProbabilityMeasure.map ⟨Measure.map (X x✝) (μ x✝), ⋯⟩ ⋯

pp: case e'_5
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝⁴ : TopologicalSpace E
F : Type u_6
inst✝³ : OpensMeasurableSpace E
inst✝² : TopologicalSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
g : E → F
hg : Continuous[inst✝⁴, inst✝²] g
h : TendstoInDistribution X l Z μ μ'
⊢ ⟨Measure.map (g ∘ Z) μ', ⋯⟩ = ProbabilityMeasure.map ⟨Measure.map Z μ', ⋯⟩ ⋯

Tactic:
·

Goal state after:
case e'_3
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝⁴ : TopologicalSpace E
F : Type u_6
inst✝³ : OpensMeasurableSpace E
inst✝² : TopologicalSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
g : E → F
hg : Continuous[inst✝⁴, inst✝²] g
h : TendstoInDistribution X l Z μ μ'
x✝ : ι
⊢ ⟨Measure.map (g ∘ X x✝) (μ x✝), ⋯⟩ = ProbabilityMeasure.map ⟨Measure.map (X x✝) (μ x✝), ⋯⟩ ⋯
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case pos
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : CompleteSpace E
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ

pp: case neg
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : ¬CompleteSpace E
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : CompleteSpace E
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case neg
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : ¬CompleteSpace E
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : ¬CompleteSpace E
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case e'_5
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝⁴ : TopologicalSpace E
F : Type u_6
inst✝³ : OpensMeasurableSpace E
inst✝² : TopologicalSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
g : E → F
hg : Continuous[inst✝⁴, inst✝²] g
h : TendstoInDistribution X l Z μ μ'
⊢ ⟨Measure.map (g ∘ Z) μ', ⋯⟩ = ProbabilityMeasure.map ⟨Measure.map Z μ', ⋯⟩ ⋯

Tactic:
·

Goal state after:
case e'_5
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝⁴ : TopologicalSpace E
F : Type u_6
inst✝³ : OpensMeasurableSpace E
inst✝² : TopologicalSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
g : E → F
hg : Continuous[inst✝⁴, inst✝²] g
h : TendstoInDistribution X l Z μ μ'
⊢ ⟨Measure.map (g ∘ Z) μ', ⋯⟩ = ProbabilityMeasure.map ⟨Measure.map Z μ', ⋯⟩ ⋯
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : ¬AEStronglyMeasurable f μ
⊢ ∫ (x : α), f x ∂μ = ∫ (x : β), g x ∂ν

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : ¬AEStronglyMeasurable f μ
⊢ ∫ (x : α), f x ∂μ = ∫ (x : β), g x ∂ν
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : p = 0
⊢ eLpNorm f p μ = eLpNorm g p ν

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
⊢ eLpNorm f p μ = eLpNorm g p ν

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : p = 0
⊢ eLpNorm f p μ = eLpNorm g p ν
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : p = ∞
⊢ eLpNorm f p μ = eLpNorm g p ν

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : ¬p = ∞
⊢ eLpNorm f p μ = eLpNorm g p ν

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : p = ∞
⊢ eLpNorm f p μ = eLpNorm g p ν
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : p = ∞
⊢ eLpNorm f p μ = eLpNorm g p ν

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : ¬p = ∞
⊢ eLpNorm f p μ = eLpNorm g p ν

Tactic:
· simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,
      if_false]
    apply essSup_eq
    exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)

Goal state after:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
h0 : ¬p = 0
h_top : ¬p = ∞
⊢ eLpNorm f p μ = eLpNorm g p ν
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: case neg
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : ¬Nonempty ι
⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε

pp: case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : ¬Nonempty ι
⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case hF_meas
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᶠ (n : ι) in l, AEMeasurable (fun a ↦ ↑(f (X n a))) μ'

pp: case h_bound
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᶠ (n : ι) in l, ∀ᵐ (a : Ω') ∂μ', ↑(f (X n a)) ≤ edist 0 f

pp: case h_fin
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∫⁻ (a : Ω'), edist 0 f ∂μ' ≠ ⊤

pp: case h_lim
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᵐ (a : Ω') ∂μ', Tendsto (fun n ↦ ↑(f (X n a))) l (𝓝 ↑(f (Z a)))

Tactic:
·

Goal state after:
case hF_meas
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᶠ (n : ι) in l, AEMeasurable (fun a ↦ ↑(f (X n a))) μ'
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ x ∈ {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} ↔ x ∈ {x | C ≤ ‖f i x‖}

Tactic:
← norm_toNNReal

Goal state after:
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ x ∈ {x | ⟨C, ⋯⟩ ≤ ‖f i x‖.toNNReal} ↔ x ∈ {x | C ≤ ‖f i x‖}
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case h_bound
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᶠ (n : ι) in l, ∀ᵐ (a : Ω') ∂μ', ↑(f (X n a)) ≤ edist 0 f

pp: case h_fin
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∫⁻ (a : Ω'), edist 0 f ∂μ' ≠ ⊤

pp: case h_lim
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᵐ (a : Ω') ∂μ', Tendsto (fun n ↦ ↑(f (X n a))) l (𝓝 ↑(f (Z a)))

Tactic:
·

Goal state after:
case h_bound
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᶠ (n : ι) in l, ∀ᵐ (a : Ω') ∂μ', ↑(f (X n a)) ≤ edist 0 f
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case h_fin
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∫⁻ (a : Ω'), edist 0 f ∂μ' ≠ ⊤

pp: case h_lim
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∀ᵐ (a : Ω') ∂μ', Tendsto (fun n ↦ ↑(f (X n a))) l (𝓝 ↑(f (Z a)))

Tactic:
·

Goal state after:
case h_fin
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
f : BoundedContinuousFunction E NNReal
⊢ ∫⁻ (a : Ω'), edist 0 f ∂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)
this : IsProbabilityMeasure μ
⊢ ∏ i, ∫ (x : 𝓧 i), f i x ∂Measure.map (X i) μ = ∏ i, ∫ (ω : Ω), f i (X i ω) ∂μ

pp: case hφ
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)
this : IsProbabilityMeasure μ
⊢ AEMeasurable (fun ω x ↦ X x ω) μ

pp: case hfm
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)
this : IsProbabilityMeasure μ
⊢ AEStronglyMeasurable (fun x ↦ ∏ i, f i (x i)) (Measure.map (fun ω x ↦ X x ω) μ)

Tactic:
·

Goal state after:
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)
this : IsProbabilityMeasure μ
⊢ ∏ i, ∫ (x : 𝓧 i), f i x ∂Measure.map (X i) μ = ∏ i, ∫ (ω : Ω), f i (X i ω) ∂μ
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
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

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_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
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.Probability.Independence.Integration

Goal state before:
pp: case hφ
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)
this : IsProbabilityMeasure μ
⊢ AEMeasurable (fun ω x ↦ X x ω) μ

pp: case hfm
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)
this : IsProbabilityMeasure μ
⊢ AEStronglyMeasurable (fun x ↦ ∏ i, f i (x i)) (Measure.map (fun ω x ↦ X x ω) μ)

Tactic:
·

Goal state after:
case hφ
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)
this : IsProbabilityMeasure μ
⊢ AEMeasurable (fun ω x ↦ X x ω) μ