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

Goal state before:
pp: case inl
Ω : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : PseudoMetrizableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
F : Set Ω
F_closed : IsClosed[inst✝³] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i ↦ (μs i) E) ⊥ (𝓝 (μ E))
⊢ limsup (fun i ↦ (μs i) F) ⊥ ≤ μ F

pp: case inr
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : PseudoMetrizableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i ↦ (μs i) E) L (𝓝 (μ E))
F : Set Ω
F_closed : IsClosed[inst✝³] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h✝ : L.NeBot
⊢ limsup (fun i ↦ (μs i) F) L ≤ μ F

Tactic:
·

Goal state after:
case inl
Ω : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : PseudoMetrizableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
F : Set Ω
F_closed : IsClosed[inst✝³] F
this : PseudoMetricSpace Ω := pseudoMetrizableSpacePseudoMetric Ω
h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i ↦ (μs i) E) ⊥ (𝓝 (μ E))
⊢ limsup (fun i ↦ (μs i) F) ⊥ ≤ μ F
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet (f '' s)

pp: case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet s

Tactic:
·

Goal state after:
case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet (f '' s)
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet s

Tactic:
·

Goal state after:
case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UniformIntegrable fn p μ
⊢ ∃ C, ∀ (i : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}), eLpNorm ((fun f ↦ ↑f) i) p μ ≤ ↑C

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UniformIntegrable fn p μ
⊢ ∃ C, ∀ (i : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}), eLpNorm ((fun f ↦ ↑f) i) p μ ≤ ↑C
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case mp
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
h : IndepFun f g κ μ
⊢ ∀ (s : Set β) (t : Set β'),
    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)

pp: case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
h :
  ∀ (s : Set β) (t : Set β'),
    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)
⊢ IndepFun f g κ μ

Tactic:
·

Goal state after:
case mp
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
h : IndepFun f g κ μ
⊢ ∀ (s : Set β) (t : Set β'),
    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)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
h :
  ∀ (s : Set β) (t : Set β'),
    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)
⊢ IndepFun f g κ μ

Tactic:
·

Goal state after:
case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
h :
  ∀ (s : Set β) (t : Set β'),
    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)
⊢ IndepFun f g κ μ
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
μ : Measure α
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
ht : t = ∅ ∨ t = univ
inst✝ : IsZeroOrMarkovKernel 0
⊢ ∀ᵐ (a : α) ∂μ, (0 a) (s ∩ t) = (0 a) s * (0 a) t

pp: case inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
ht : t = ∅ ∨ t = univ
h : IsMarkovKernel κ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (s ∩ t) = (κ a) s * (κ a) t

Tactic:
·

Goal state after:
case inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
μ : Measure α
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
ht : t = ∅ ∨ t = univ
inst✝ : IsZeroOrMarkovKernel 0
⊢ ∀ᵐ (a : α) ∂μ, (0 a) (s ∩ t) = (0 a) s * (0 a) t
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ ↑↑μ (s ∩ range f) = ↑↑μ s

pp: case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet (f '' s)

pp: case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ ↑↑μ (s ∩ range f) = ↑↑μ s
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = ∅
⊢ (κ a) (s ∩ t) = (κ a) s * (κ a) t

pp: case inr.inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = univ
⊢ (κ a) (s ∩ t) = (κ a) s * (κ a) t

Tactic:
·

Goal state after:
case inr.inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = ∅
⊢ (κ a) (s ∩ t) = (κ a) s * (κ a) t
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = univ
⊢ (κ a) (s ∩ t) = (κ a) s * (κ a) t

Tactic:
·

Goal state after:
case inr.inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m' _mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
s t : Set Ω
a✝ : s ∈ {s | MeasurableSet s}
h : IsMarkovKernel κ
a : α
ht : t = univ
⊢ (κ a) (s ∩ t) = (κ a) s * (κ a) t
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet (f '' s)

pp: case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet (f '' s)
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : ↑M
s : Set Ω'
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s₁ s₂ s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h₁ : IndepSets s₁ s' κ μ
h₂ : IndepSets s₂ s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ s₁ ∪ s₂
ht2 : t2 ∈ s'
ht1₁ : t1 ∈ s₁
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

pp: case inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s₁ s₂ s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h₁ : IndepSets s₁ s' κ μ
h₂ : IndepSets s₂ s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ s₁ ∪ s₂
ht2 : t2 ∈ s'
ht1₂ : t1 ∈ s₂
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
·

Goal state after:
case inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s₁ s₂ s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h₁ : IndepSets s₁ s' κ μ
h₂ : IndepSets s₂ s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ s₁ ∪ s₂
ht2 : t2 ∈ s'
ht1₁ : t1 ∈ s₁
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s₁ s₂ s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h₁ : IndepSets s₁ s' κ μ
h₂ : IndepSets s₂ s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ s₁ ∪ s₂
ht2 : t2 ∈ s'
ht1₂ : t1 ∈ s₂
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
·

Goal state after:
case inr
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s₁ s₂ s' : Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h₁ : IndepSets s₁ s' κ μ
h₂ : IndepSets s₂ s' κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ s₁ ∪ s₂
ht2 : t2 ∈ s'
ht1₂ : t1 ∈ s₂
⊢ ∀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
t1 t2 : Set Ω
ht1 : t1 ∈ ⋂ n, s n
ht2 : t2 ∈ s'
n : ι
h : IndepSets (s n) s' κ μ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
exact h t1 t2 (Set.mem_iInter.mp ht1 n) ht2

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case calc_1
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
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} ≤ ∫⁻ (t : ℝ) in Ioi 0, liminf (fun i ↦ (μs i) {a | t < f a}) atTop

pp: case calc_2
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
⊢ ∫⁻ (t : ℝ) in Ioi 0, liminf (fun i ↦ (μs i) {a | t < f a}) atTop ≤
    liminf (fun i ↦ ∫⁻ (t : ℝ) in Ioi 0, (μs i) {a | t < f a}) atTop

Tactic:
·

Goal state after:
case calc_1
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
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} ≤ ∫⁻ (t : ℝ) in Ioi 0, liminf (fun i ↦ (μs i) {a | t < f a}) atTop
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case calc_2
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
⊢ ∫⁻ (t : ℝ) in Ioi 0, liminf (fun i ↦ (μs i) {a | t < f a}) atTop ≤
    liminf (fun i ↦ ∫⁻ (t : ℝ) in Ioi 0, (μs i) {a | t < f a}) atTop

Tactic:
·

Goal state after:
case calc_2
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
⊢ ∫⁻ (t : ℝ) in Ioi 0, liminf (fun i ↦ (μs i) {a | t < f a}) atTop ≤
    liminf (fun i ↦ ∫⁻ (t : ℝ) in Ioi 0, (μs i) {a | t < f a}) atTop
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
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x = i
⊢ (if x = i then t₁ else t₂) ∈ s x

pp: case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x ∈ {j}
⊢ (if x = i then t₁ else t₂) ∈ s x

Tactic:
·

Goal state after:
case inl
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x = i
⊢ (if x = i then t₁ else t₂) ∈ s x
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x ∈ {j}
⊢ (if x = i then t₁ else t₂) ∈ s x

Tactic:
·

Goal state after:
case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx✝ : x ∈ {i, j}
hx : x ∈ {j}
⊢ (if x = i then t₁ else t₂) ∈ s x
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
μ : Measure α
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
ht2m : MeasurableSet t2
inst✝ : IsZeroOrMarkovKernel 0
hyp : IndepSets p1 p2 0 μ
⊢ ∀ᵐ (a : α) ∂μ, (0 a) (t1 ∩ t2) = (0 a) t1 * (0 a) t2

pp: case inr
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
ht2m : MeasurableSet t2
h : IsMarkovKernel κ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
·

Goal state after:
case inl
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
μ : Measure α
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
ht2m : MeasurableSet t2
inst✝ : IsZeroOrMarkovKernel 0
hyp : IndepSets p1 p2 0 μ
⊢ ∀ᵐ (a : α) ∂μ, (0 a) (t1 ∩ t2) = (0 a) t1 * (0 a) t2
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
β : Type u_6
β' : Type u_7
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
⊢ f ⟂ᵢ[μ] g ↔
    ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t)

Tactic:
by
  simp only [IndepFun, Kernel.indepFun_iff_measure_inter_preimage_eq_mul, ae_dirac_eq,
    Filter.eventually_pure, Kernel.const_apply]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
⊢ liminf (fun i ↦ ∫ (x : Ω), f x ∂μs i) atTop = (liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop).toReal

pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
⊢ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop ≠ ∞

Tactic:
·

Goal state after:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
⊢ liminf (fun i ↦ ∫ (x : Ω), f x ∂μs i) atTop = (liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop).toReal