mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: case refine_1
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
γ : Type u_2
𝕜 : Type u_3
inst✝ : RCLike 𝕜
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i ↦ ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ 𝕜
⊢ Tendsto (fun i ↦ ∫ (ω : Ω), f ω ∂↑(μs i)) F
    (𝓝 (↑(∫ (x : Ω), RCLike.re (f x) ∂↑μ) + ↑(∫ (x : Ω), RCLike.im (f x) ∂↑μ) * RCLike.I))

Tactic:
[

Goal state after:
case refine_1
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
γ : Type u_2
𝕜 : Type u_3
inst✝ : RCLike 𝕜
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i ↦ ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ 𝕜
⊢ Tendsto (fun i ↦ ∫ (ω : Ω), f ω ∂↑(μs i)) F
    (𝓝 (↑(∫ (x : Ω), RCLike.re (f x) ∂↑μ) + ↑(∫ (x : Ω), RCLike.im (f x) ∂↑μ) * RCLike.I))
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i ↦ (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ μ Eᶜ ≤ liminf (fun i ↦ (μs i) Eᶜ) L

Tactic:
[

Goal state after:
case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i ↦ (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ μ Eᶜ ≤ liminf (fun i ↦ (μs i) Eᶜ) L
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal ↑δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
⊢ ↑C * ↑δ ^ (1 / p.toReal) ≤ ↑C * μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / p.toReal)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal ↑δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
⊢ ↑C * ↑δ ^ (1 / p.toReal) ≤ ↑C * μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i ↦ (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ limsup (fun i ↦ (μs i) Eᶜ) L ≤ μ Eᶜ

Tactic:
[

Goal state after:
case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i ↦ (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ limsup (fun i ↦ (μs i) Eᶜ) L ≤ μ Eᶜ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator ((↑n)⁻¹ • ∑ i ∈ Finset.range n, f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
[

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator ((↑n)⁻¹ • ∑ i ∈ Finset.range n, f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), s.indicator ((↑n)⁻¹ • f i) = (↑n)⁻¹ • s.indicator (f i)
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm (s.indicator ((↑n)⁻¹ • f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
[

Goal state after:
case refine_2.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), s.indicator ((↑n)⁻¹ • f i) = (↑n)⁻¹ • s.indicator (f i)
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm (s.indicator ((↑n)⁻¹ • f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
⊢ ∃ C, ∀ (i : ℕ), eLpNorm ((↑i)⁻¹ • ∑ i ∈ Finset.range i, f i) p μ ≤ ↑C

Tactic:
[

Goal state after:
case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
⊢ ∃ C, ∀ (i : ℕ), eLpNorm ((↑i)⁻¹ • ∑ i ∈ Finset.range i, f i) p μ ≤ ↑C
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_3.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
n : ℕ
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm ((↑n)⁻¹ • f i) p μ ≤ ↑C

Tactic:
[

Goal state after:
case refine_3.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
n : ℕ
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm ((↑n)⁻¹ • f i) p μ ≤ ↑C
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: 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
π : ι → Set (Set Ω)
ι' : Type u_5
g : ι' → ι
hg : Function.Injective g
h : iIndepSets π κ μ
s : Finset ι'
f : ι' → Set Ω
hf : ∀ i ∈ s, f i ∈ (π ∘ g) i
f' : ι → Set Ω := Function.extend g f fun x ↦ ∅
f'_apply : ∀ (x : ι'), f' (g x) = f x
⊢ ∀ i ∈ Finset.image g s, f' i ∈ π i

Tactic:
[

Goal state after:
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
π : ι → Set (Set Ω)
ι' : Type u_5
g : ι' → ι
hg : Function.Injective g
h : iIndepSets π κ μ
s : Finset ι'
f : ι' → Set Ω
hf : ∀ i ∈ s, f i ∈ (π ∘ g) i
f' : ι → Set Ω := Function.extend g f fun x ↦ ∅
f'_apply : ∀ (x : ι'), f' (g x) = f x
⊢ ∀ i ∈ Finset.image g s, f' i ∈ π i
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x ↦ {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i ↦ if hi_mem : i ∈ S then Exists.choose ⋯ else univ
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (setsβ i)

Tactic:
[

Goal state after:
α : Type u_1
Ω : Type u_2
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x ↦ {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i ↦ if hi_mem : i ∈ S then Exists.choose ⋯ else univ
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (setsβ i)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x ↦ {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i ↦ if hi_mem : i ∈ S then Exists.choose ⋯ else univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
i : ι
hi_mem : i ∈ S
⊢ setsΩ i = f i ⁻¹' setsβ i

Tactic:
[

Goal state after:
α : Type u_1
Ω : Type u_2
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x ↦ {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i ↦ if hi_mem : i ∈ S then Exists.choose ⋯ else univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
i : ι
hi_mem : i ∈ S
⊢ setsΩ i = f i ⁻¹' setsβ i
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
u : Set ι
hyp : ∀ n ∈ u, IndepSets (s n) s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ ⋃ n ∈ u, s n
ht2 : t2 ∈ s'
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
[

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
u : Set ι
hyp : ∀ n ∈ u, IndepSets (s n) s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ ⋃ n ∈ u, s n
ht2 : t2 ∈ s'
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
u : Set ι
h : ∃ n ∈ u, IndepSets (s n) s' κ μ
⊢ IndepSets (⋂ n ∈ u, s n) s' κ μ

Tactic:
by
  intro t1 t2 ht1 ht2
  rcases h with ⟨n, hn, h⟩
  exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
f : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
⊢ iIndep (fun i ↦ MeasurableSpace.comap (fun x ↦ x ∈ f i) ⊤) κ μ ↔ iIndepSet f κ μ

Tactic:
[

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
f : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
⊢ iIndep (fun i ↦ MeasurableSpace.comap (fun x ↦ x ∈ f i) ⊤) κ μ ↔ iIndepSet f κ μ
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous[inst✝¹, _] f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝¹] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
⊢ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop

Tactic:
[

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous[inst✝¹, _] f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝¹] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
⊢ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
s : ι → Set Ω
⊢ iIndepSets (fun i ↦ {s i}) μ ↔ ∀ (t : Finset ι), μ (⋂ i ∈ t, s i) = ∏ i ∈ t, μ (s i)

Tactic:
[

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
s : ι → Set Ω
⊢ iIndepSets (fun i ↦ {s i}) μ ↔ ∀ (t : Finset ι), μ (⋂ i ∈ t, s i) = ∏ i ∈ t, μ (s i)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : ι → Set Ω
hf : ∀ (i : ι), MeasurableSet (f i)
⊢ iIndepSet f μ ↔ ∀ (s : Finset ι), μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)

Tactic:
[

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : ι → Set Ω
hf : ∀ (i : ι), MeasurableSet (f i)
⊢ iIndepSet f μ ↔ ∀ (s : Finset ι), μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.basic
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : u ∈ p2
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ u) = (κ a) t1 * (κ a) u

Tactic:
exact hyp t1 u ht1 hu

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

Goal state before:
pp: 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
γ : ι → Type u_9
mβ : (i : ι) → MeasurableSpace (β i)
mγ : (i : ι) → MeasurableSpace (γ i)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
g : (i : ι) → β i → γ i
hg : ∀ (i : ι), Measurable (g i)
t : Finset ι
s : (i : ι) → Set (γ i)
hs : ∀ i ∈ t, MeasurableSet (s i)
this : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ t, f i ⁻¹' g i ⁻¹' s i) = ∏ i ∈ t, (κ a) (f i ⁻¹' g i ⁻¹' s i)
a : α
ha : (κ a) (⋂ i ∈ t, f i ⁻¹' g i ⁻¹' s i) = ∏ i ∈ t, (κ a) (f i ⁻¹' g i ⁻¹' s i)
⊢ (κ a) (⋂ i ∈ t, g i ∘ f i ⁻¹' s i) = ∏ i ∈ t, (κ a) (g i ∘ f i ⁻¹' s i)

Tactic:
[

Goal state after:
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
γ : ι → Type u_9
mβ : (i : ι) → MeasurableSpace (β i)
mγ : (i : ι) → MeasurableSpace (γ i)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
g : (i : ι) → β i → γ i
hg : ∀ (i : ι), Measurable (g i)
t : Finset ι
s : (i : ι) → Set (γ i)
hs : ∀ i ∈ t, MeasurableSet (s i)
this : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ t, f i ⁻¹' g i ⁻¹' s i) = ∏ i ∈ t, (κ a) (f i ⁻¹' g i ⁻¹' s i)
a : α
ha : (κ a) (⋂ i ∈ t, f i ⁻¹' g i ⁻¹' s i) = ∏ i ∈ t, (κ a) (f i ⁻¹' g i ⁻¹' s i)
⊢ (κ a) (⋂ i ∈ t, g i ∘ f i ⁻¹' s i) = ∏ i ∈ t, (κ a) (g i ∘ f i ⁻¹' s i)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f✝ : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ
s : Finset ι
f : ι → Set Ω
hs : ∀ i ∈ s, MeasurableSet (f i)
this : ⋂ i ∈ s, f i = ⋂ i, f ↑i
⊢ μ (⋂ i ∈ s, f i) = ∏ i, μ (f ↑i)

Tactic:
this

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f✝ : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ
s : Finset ι
f : ι → Set Ω
hs : ∀ i ∈ s, MeasurableSet (f i)
this : ⋂ i ∈ s, f i = ⋂ i, f ↑i
⊢ μ (⋂ i, f ↑i) = ∏ i, μ (f ↑i)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_1
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
h_set_eq : ∀ (ω : α), Prod.mk ω ⁻¹' u ×ˢ s ×ˢ t = if ω ∈ u then s ×ˢ t else ∅
⊢ (fun a ↦ ((κ.map fun ω ↦ (f ω, g ω)) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t)) =ᵐ[μ] fun a ↦
    ((κ.map f ×ₖ κ.map g) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t)

Tactic:
[

Goal state after:
case refine_1
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
h_set_eq : ∀ (ω : α), Prod.mk ω ⁻¹' u ×ˢ s ×ˢ t = if ω ∈ u then s ×ˢ t else ∅
⊢ (fun a ↦ ((κ.map fun ω ↦ (f ω, g ω)) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t)) =ᵐ[μ] fun a ↦
    ((κ.map f ×ₖ κ.map g) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case h
Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h₁ : IsTightMeasureSet (range (ProbabilityMeasure.toMeasure ∘ μs))
h₂ : ∀ (F : Set Ω), IsCompact F → limsup (fun x ↦ (μs x) F) L ≤ μ F
h✝ : L.NeBot
F : Set Ω
hF_closed : IsClosed[inst✝²] F
ε : ℝ≥0
hε : 0 < ε
x✝ : ↑(μ F) < ∞
K : Set Ω
hKc : IsCompact K
hK_le : ∀ μ ∈ range (ProbabilityMeasure.toMeasure ∘ μs), μ Kᶜ ≤ ↑ε
i : ι
⊢ ↑((μs i) F) ≤ ↑((μs i) (F ∩ K)) + ↑ε

Tactic:
[

Goal state after:
case h
Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h₁ : IsTightMeasureSet (range (ProbabilityMeasure.toMeasure ∘ μs))
h₂ : ∀ (F : Set Ω), IsCompact F → limsup (fun x ↦ (μs x) F) L ≤ μ F
h✝ : L.NeBot
F : Set Ω
hF_closed : IsClosed[inst✝²] F
ε : ℝ≥0
hε : 0 < ε
x✝ : ↑(μ F) < ∞
K : Set Ω
hKc : IsCompact K
hK_le : ∀ μ ∈ range (ProbabilityMeasure.toMeasure ∘ μs), μ Kᶜ ≤ ↑ε
i : ι
⊢ ↑((μs i) F) ≤ ↑((μs i) (F ∩ K)) + ↑ε
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
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
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
i : ι
hi1 : i ∈ p1
hi2 : i ∉ p2
⊢ g i ∈ s i

Tactic:
[

Goal state after:
case inl
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
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
i : ι
hi1 : i ∈ p1
hi2 : i ∉ p2
⊢ g i ∈ s i
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
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
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
i : ι
hi1 : i ∈ p1
hi2 : i ∉ p2
⊢ g i ∈ s i

Tactic:
simp_rw [g, if_pos hi1, if_neg hi2, Set.inter_univ]

Goal state after:
case inl
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
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
i : ι
hi1 : i ∈ p1
hi2 : i ∉ p2
⊢ f1 i ∈ s i