mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: β : Type u_1
Ω : Type u_2
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ : Measure (β × Ω)
inst✝² : IsFiniteMeasure ρ
E : Type u_3
f : β × Ω → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t
hf : IntegrableOn f (s ×ˢ t) ρ
⊢ ∫ (b : β) in s, ∫ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in s ×ˢ t, f x ∂ρ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun m ↦ κ.densityProcess ν n a x (seq m)) atTop (𝓝 (κ.densityProcess ν n a x ∅))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: β : Type u_1
Ω : Type u_2
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ : Measure (β × Ω)
inst✝² : IsFiniteMeasure ρ
E : Type u_3
f : β × Ω → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set β
hs : MeasurableSet s
hf : IntegrableOn f (s ×ˢ Set.univ) ρ
⊢ ∫ (b : β) in s, ∫ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in s ×ˢ Set.univ, f x ∂ρ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto (fun m ↦ κ.densityProcess ν n a x (seq m)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : γ) ∂ν a,
    Tendsto (fun n ↦ κ.densityProcess ν n a x s) atTop
      (𝓝 (Filtration.limitProcess (fun n x ↦ κ.densityProcess ν n a x s) (countableFiltration γ) (ν a) x))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: β : Type u_1
Ω : Type u_2
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ : Measure (β × Ω)
inst✝² : IsFiniteMeasure ρ
E : Type u_3
f : β × Ω → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : Set Ω
ht : MeasurableSet t
hf : IntegrableOn f (Set.univ ×ˢ t) ρ
⊢ ∫ (b : β), ∫ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in Set.univ ×ˢ t, f x ∂ρ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ MemLp (Filtration.limitProcess (fun n x ↦ κ.densityProcess ν n a x s) (countableFiltration γ) (ν a)) 1 (ν a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: α : Type u_1
Ω : Type u_2
F : Type u_4
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf : AEStronglyMeasurable f ρ
⊢ (∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)) ∧
      Integrable (fun a ↦ ∫ (ω : Ω), ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔
    Integrable f ρ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
this : {x | 2 ≤ ‖κ.densityProcess ν n a x s‖₊} = ∅
⊢ eLpNorm ({x | 2 ≤ ‖κ.densityProcess ν n a x s‖₊}.indicator fun x ↦ κ.densityProcess ν n a x s) 1 (ν a) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: α : Type u_1
Ω : Type u_2
F : Type u_4
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf_int : Integrable f ρ
⊢ ∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Unique

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : Kernel α ℝ
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: α : Type u_1
Ω : Type u_2
F : Type u_4
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf_int : Integrable f ρ
⊢ Integrable (fun x ↦ ∫ (y : Ω), ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: α : Type u_1
Ω : Type u_2
E : Type u_3
mα : MeasurableSpace α
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → E
hf_int : Integrable f ρ
⊢ Integrable (fun x ↦ ‖∫ (y : Ω), f (x, y) ∂ρ.condKernel x‖) ρ.fst

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Tendsto
    (fun n ↦
      eLpNorm
        ((fun x ↦ κ.densityProcess ν n a x s) -
          Filtration.limitProcess (fun n x ↦ κ.densityProcess ν n a x s) (countableFiltration γ) (ν a))
        1 (ν a))
    atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ (fun x ↦ κ.density ν a x s) =ᵐ[ν a]
    Filtration.limitProcess (fun n x ↦ κ.densityProcess ν n a x s) (countableFiltration γ) (ν a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
inst✝² : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
inst✝ : MeasurableSpace β
⊢ IsMarkovKernel (condDistrib Y X μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass β
hY : Measurable Y
x : β
hX : (Measure.map X μ) {x} ≠ 0
s : Set Ω
⊢ ((condDistrib Y X μ) x) s = ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a ↦ (X a, Y a)) μ) ({x} ×ˢ s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
inst✝ : IsFiniteKernel ν
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : γ) ∂ν a, Tendsto (fun n ↦ κ.densityProcess ν n a x s) atTop (𝓝 (κ.density ν a x s))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
x : γ
s : Set β
hs : MeasurableSet s
⊢ Measurable fun a ↦ κ.density ν a x s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hY : AEMeasurable Y μ
⊢ Measure.map X μ ⊗ₘ condDistrib Y X μ = Measure.map (fun a ↦ (X a, Y a)) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
s : Set β
hs : MeasurableSet s
a : α
⊢ Measurable fun x ↦ κ.density ν a x s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
⊢ ⇑(condDistrib Y X μ) ∘ₘ Measure.map X μ = Measure.map Y μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Unique

Goal state before:
pp: α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
ρ : Measure (α × Ω)
inst✝¹ : IsFiniteMeasure ρ
κ : Kernel α Ω
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
f : Ω → ℝ := embeddingReal Ω
hf : MeasurableEmbedding (embeddingReal Ω)
ρ' : Measure (α × ℝ) := Measure.map (Prod.map id f) ρ
hρ'def : ρ' = Measure.map (Prod.map id f) ρ
⊢ ρ'.fst = ρ.fst

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
x : γ
s s' : Set β
h : s ⊆ s'
⊢ κ.density ν a x s ≤ κ.density ν a x s'

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
X' : α → β
Y' : α → Ω
hY : Y =ᵐ[μ] Y'
hX : X =ᵐ[μ] X'
a : α
ha : X a = X' a
hb : Y a = Y' a
⊢ (X a, Y a) = (X' a, Y' a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
X' : α → β
Y' : α → Ω
hY : Y =ᵐ[μ] Y'
hX : X =ᵐ[μ] X'
⊢ condDistrib Y X μ = condDistrib Y' X' μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
X' : α → β
hX : X =ᵐ[μ] X'
⊢ Y =ᵐ[μ] Y

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
x : γ
s : Set β
⊢ 0 ≤ κ.density ν a x s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
Y' : α → Ω
hY : Y =ᵐ[μ] Y'
⊢ X =ᵐ[μ] X

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
x : γ
s : Set β
⊢ κ.density ν a x s ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
s : Set β
⊢ eLpNorm (fun x ↦ κ.density ν a x s) 1 (ν a) ≤ (ν a) univ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map (fun a ↦ (X a, Y a)) μ)
⊢ (∀ᵐ (a : β) ∂Measure.map X μ, Integrable (fun ω ↦ f (a, ω)) ((condDistrib Y X μ) a)) ∧
      Integrable (fun a ↦ ∫ (ω : Ω), ‖f (a, ω)‖ ∂(condDistrib Y X μ) a) (Measure.map X μ) ↔
    Integrable f (Measure.map (fun a ↦ (X a, Y a)) μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun x ↦ κ.density ν a x s) (ν a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
inst✝² : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝ : NormedSpace ℝ F
hf : StronglyMeasurable f
⊢ StronglyMeasurable fun x ↦ ∫ (y : Ω), f (x, y) ∂(condDistrib Y X μ) x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
⊢ Tendsto (fun i ↦ ∫ (x : γ) in A, κ.densityProcess ν i a x s ∂ν a) atTop (𝓝 (∫ (x : γ) in A, κ.density ν a x s ∂ν a))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ∫ (x : γ) in A, κ.densityProcess ν n a x s ∂ν a
⊢ ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
⊢ κ.fst ⊗ₖ borelMarkovFromReal Ω η = ((κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η).comapRight ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
inst✝² : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝ : NormedSpace ℝ F
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map (fun a ↦ (X a, Y a)) μ)
⊢ AEStronglyMeasurable (fun x ↦ ∫ (y : Ω), f (x, y) ∂(condDistrib Y X μ) x) (Measure.map X μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Unique

Goal state before:
pp: α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
ρ : Measure (α × Ω)
inst✝¹ : IsFiniteMeasure ρ
κ : Kernel α Ω
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
f : Ω → ℝ := embeddingReal Ω
hf : MeasurableEmbedding (embeddingReal Ω)
ρ' : Measure (α × ℝ) := Measure.map (Prod.map id f) ρ
hρ'def : ρ' = Measure.map (Prod.map id f) ρ
hρ' : ρ'.fst = ρ.fst
⊢ ∀ᵐ (x : α) ∂ρ.fst, (κ.map f) x = ρ'.condKernel x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
c : α
t : Set (β × Ω)
a✝ : MeasurableSet t
⊢ Measurable (Prod.map id e)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x ↦ (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
⊢ Measure.map X μ = (Measure.map (fun x ↦ (X x, Y x)) μ).fst

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
⊢ κ = κ'.comapRight h_prod_embed

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = limsup (fun i ↦ ∫ (x : γ) in A, κ.densityProcess ν i a x s ∂ν a) atTop
⊢ ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ∫ (x : γ) in A, κ.densityProcess ν n a x s ∂ν a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x ↦ (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
⊢ ⇑(condDistrib Y X μ) =ᵐ[Measure.map X μ] ⇑κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
⊢ ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫ (x : γ), κ.density ν a x s ∂ν a = (κ a).real (univ ×ˢ s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
X : α → β
hY : AEMeasurable Y μ
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x ↦ (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
hX : AEMeasurable X μ
this : ⇑(condDistrib (AEMeasurable.mk Y hY) (AEMeasurable.mk X hX) μ) =ᵐ[Measure.map (AEMeasurable.mk X hX) μ] ⇑κ
⊢ ⇑(condDistrib Y X μ) =ᵐ[Measure.map X μ] ⇑κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
⊢ κ.fst ⊗ₖ borelMarkovFromReal Ω η = κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ mγ = ⨆ n, ↑(countableFiltration γ) n

Tactic:
iSup_countableFiltration

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ mγ = mγ