mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ : Measure α
κ : Kernel α β
f : β → γ
hf : Measurable f
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : α), (κ a) (f ⁻¹' s) ∂μ = ∫⁻ (a : α), ((κ.map f) a) s ∂μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
f : β → γ
hf : Measurable f
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : α), (κ a) (f ⁻¹' s) ∂μ = ∫⁻ (a : α), ((κ.map f) a) s ∂μ
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
⊢ map Function.diag (map f μ) = map (Function.prod f f) μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
⊢ map Function.diag (map f μ) = map (Function.prod f f) μ
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝¹ : SFinite μ
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
⊢ (Kernel.id ×ₖ κ ∘ₖ Kernel.const Unit μ) () = ⇑(Kernel.id ×ₖ κ) ∘ₘ μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝¹ : SFinite μ
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
⊢ (Kernel.id ×ₖ κ ∘ₖ Kernel.const Unit μ) () = ⇑(Kernel.id ×ₖ κ) ∘ₘ μ
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel α γ
inst✝ : IsSFiniteKernel η
⊢ η ×ₖ κ = η ×ₖ κ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ map Function.diag μ = map Function.diag μ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
a : α
κ : Kernel α β
η : Kernel (α × β) γ
p : β × γ → Prop
hp : MeasurableSet {x | p x}
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
h : (κ a) {a_1 | ¬(η (a, a_1)) {a | ¬p (a_1, a)} = 0} = 0
⊢ (fun b ↦ (η (a, b)) (Prod.mk b ⁻¹' {a | ¬p a})) =ᵐ[κ a] 0

pp: case pos.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
a : α
κ : Kernel α β
η : Kernel (α × β) γ
p : β × γ → Prop
hp : MeasurableSet {x | p x}
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
h : (κ a) {a_1 | ¬(η (a, a_1)) {a | ¬p (a_1, a)} = 0} = 0
⊢ MeasurableSet {a | ¬p a}

Tactic:
]

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
a : α
κ : Kernel α β
η : Kernel (α × β) γ
p : β × γ → Prop
hp : MeasurableSet {x | p x}
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
h : (κ a) {a_1 | ¬(η (a, a_1)) {a | ¬p (a_1, a)} = 0} = 0
⊢ (fun b ↦ (η (a, b)) (Prod.mk b ⁻¹' {a | ¬p a})) =ᵐ[κ a] 0

case pos.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
a : α
κ : Kernel α β
η : Kernel (α × β) γ
p : β × γ → Prop
hp : MeasurableSet {x | p x}
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
h : (κ a) {a_1 | ¬(η (a, a_1)) {a | ¬p (a_1, a)} = 0} = 0
⊢ MeasurableSet {a | ¬p a}
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ Kernel.id = deterministic (fun a ↦ (a.1, a.2)) ⋯

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ Kernel.id = deterministic (fun a ↦ (a.1, a.2)) ⋯
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : α), ∫⁻ (b : β), (η (a, b)) s ∂κ a ∂μ = ∫⁻ (a : α), ((κ ⊗ₖ η) a) (Prod.snd ⁻¹' s) ∂μ

pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
⊢ MeasurableSet (Prod.snd ⁻¹' s)

Tactic:
]

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : α), ∫⁻ (b : β), (η (a, b)) s ∂κ a ∂μ = ∫⁻ (a : α), ((κ ⊗ₖ η) a) (Prod.snd ⁻¹' s) ∂μ

α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
⊢ MeasurableSet (Prod.snd ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
a : α
⊢ (κ a).prod ((ξ a).prod (η a)) = (κ a).prod ((ξ a).prod (η a))

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
a : α
u : Set (β × γ)
hu : MeasurableSet u
⊢ ∫⁻ (b : β), ((η.restrict ht) (a, b)) (Prod.mk b ⁻¹' u) ∂(κ.restrict hs) a =
    ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' (u ∩ s ×ˢ t)) ∂κ a

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
a : α
u : Set (β × γ)
hu : MeasurableSet u
⊢ ∫⁻ (b : β), ((η.restrict ht) (a, b)) (Prod.mk b ⁻¹' u) ∂(κ.restrict hs) a =
    ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' (u ∩ s ×ˢ t)) ∂κ a
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
⊢ (κ ×ₖ ξ ×ₖ η).map (⇑MeasurableEquiv.prodAssoc.symm ∘ ⇑MeasurableEquiv.prodAssoc) = κ ×ₖ ξ ×ₖ η

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
⊢ (κ ×ₖ ξ ×ₖ η).map (⇑MeasurableEquiv.prodAssoc.symm ∘ ⇑MeasurableEquiv.prodAssoc) = κ ×ₖ ξ ×ₖ η
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: case pos.e_f
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ ∫⁻ (b : β), (η (a, b)) s ∂κ a = ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Prod.snd ⁻¹' s) ∂κ a

pp: case pos.e_f.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ MeasurableSet (Prod.snd ⁻¹' s)

Tactic:
]

Goal state after:
case pos.e_f
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ ∫⁻ (b : β), (η (a, b)) s ∂κ a = ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Prod.snd ⁻¹' s) ∂κ a

case pos.e_f.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ MeasurableSet (Prod.snd ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
a : α
u : Set (β × γ)
hu : MeasurableSet u
b : β
⊢ (η (a, b)) {c | (b, c) ∈ u ∧ b ∈ s ∧ c ∈ t} = if b ∈ s then (η (a, b)) ({c | (b, c) ∈ u} ∩ t) else 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
a : α
u : Set (β × γ)
hu : MeasurableSet u
b : β
⊢ (η (a, b)) {c | (b, c) ∈ u ∧ b ∈ s ∧ c ∈ t} = if b ∈ s then (η (a, b)) ({c | (b, c) ∈ u} ∩ t) else 0
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ : Measure α
κ : Kernel α β
η : Kernel β γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel κ
⊢ ⇑(η.comap Prod.snd ⋯) ∘ₘ μ ⊗ₘ κ = ⇑(η.comap Prod.snd ⋯) ∘ₘ μ ⊗ₘ κ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝ : SFinite μ
f : α → β
hf : Measurable f
⊢ map (fun a ↦ (id a, f a)) μ = map (fun a ↦ (a, f a)) μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝ : SFinite μ
f : α → β
hf : Measurable f
⊢ map (fun a ↦ (id a, f a)) μ = map (fun a ↦ (a, f a)) μ
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
a : α
u : Set (β × γ)
hu : MeasurableSet u
this :
  ∀ (b : β), (η (a, b)) {c | (b, c) ∈ u ∧ b ∈ s ∧ c ∈ t} = s.indicator (fun b ↦ (η (a, b)) ({c | (b, c) ∈ u} ∩ t)) b
⊢ ∫⁻ (b : β) in s, (η (a, b)) ({x | (b, x) ∈ u} ∩ t) ∂κ a =
    ∫⁻ (a_1 : β) in s, (η (a, a_1)) ({c | (a_1, c) ∈ u} ∩ t) ∂κ a

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
hs : MeasurableSet s
⊢ κ.restrict hs ⊗ₖ η = κ.restrict hs ⊗ₖ η.restrict ⋯

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
hs : MeasurableSet s
⊢ κ.restrict hs ⊗ₖ η = κ.restrict hs ⊗ₖ η.restrict ⋯
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
t : Set γ
ht : MeasurableSet t
⊢ κ ⊗ₖ η.restrict ht = κ.restrict ⋯ ⊗ₖ η.restrict ht

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
t : Set γ
ht : MeasurableSet t
⊢ κ ⊗ₖ η.restrict ht = κ.restrict ⋯ ⊗ₖ η.restrict ht
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
⊢ ⨆ n, ∫⁻ (a : β × γ), (F n) a ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) ∂κ a

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
⊢ ⨆ n, ∫⁻ (a : β × γ), (F n) a ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) ∂κ a
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
⊢ ⇑κ ∘ₘ μ + ⇑η ∘ₘ μ = ⇑κ ∘ₘ μ + ⇑η ∘ₘ μ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a✝ : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
a : β
⊢ ⨆ n, ∫⁻ (a_1 : γ), (F n) (a, a_1) ∂η (a✝, a) = ⨆ n, ∫⁻ (c : γ), (F n) (a, c) ∂η (a✝, a)

pp: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a✝ : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
a : β
⊢ ∀ (n : ℕ), Measurable fun c ↦ (F n) (a, c)

pp: case h_mono
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a✝ : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
a : β
⊢ Monotone fun n c ↦ (F n) (a, c)

Tactic:
]

Goal state after:
case hf
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a✝ : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
a : β
⊢ ∀ (n : ℕ), Measurable fun c ↦ (F n) (a, c)

case h_mono
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a✝ : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
a : β
⊢ Monotone fun n c ↦ (F n) (a, c)
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
μ ν : Measure α
hμν : μ ≪ ν
κ : Kernel α γ
s : Set γ
hs : MeasurableSet s
hs_zero : (fun a ↦ (κ a) s) =ᵐ[ν] 0
⊢ (fun a ↦ (κ a) s) =ᵐ[μ] 0

Tactic:
]

Goal state after:
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
μ ν : Measure α
hμν : μ ≪ ν
κ : Kernel α γ
s : Set γ
hs : MeasurableSet s
hs_zero : (fun a ↦ (κ a) s) =ᵐ[ν] 0
⊢ (fun a ↦ (κ a) s) =ᵐ[μ] 0
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : CompleteSpace E
⊢ StronglyMeasurable fun x ↦ ∫ (y : β), f y ∂κ x

Tactic:
borelize E

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ StronglyMeasurable fun x ↦ ∫ (y : β), f y ∂κ x
Module: Mathlib.Probability.Kernel.Composition.CompProd

Goal state before:
pp: case h_mono
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a✝ : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
a : β
⊢ Monotone fun n c ↦ (F n) (a, c)

Tactic:
· exact fun i j hij b => h_mono hij _

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
μ : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ≪ η a
s : Set β
hs : MeasurableSet s
hs_zero : (fun a ↦ (η a) s) =ᵐ[μ] 0
⊢ (fun a ↦ (κ a) s) =ᵐ[μ] 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
μ : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ≪ η a
s : Set β
hs : MeasurableSet s
hs_zero : (fun a ↦ (η a) s) =ᵐ[μ] 0
⊢ (fun a ↦ (κ a) s) =ᵐ[μ] 0
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
⊢ ∀ (x : α), μ {x} ≠ 0 → κ x ≪ sum fun ω ↦ μ {ω} • κ ω

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
⊢ ∀ (x : α), μ {x} ≠ 0 → κ x ≪ sum fun ω ↦ μ {ω} • κ ω
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
b : Bool
⊢ ⇑κ ∘ₘ (boolKernel μ ν) b = (boolKernel (⇑κ ∘ₘ μ) (⇑κ ∘ₘ ν)) b

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
b : Bool
⊢ ⇑κ ∘ₘ (boolKernel μ ν) b = (boolKernel (⇑κ ∘ₘ μ) (⇑κ ∘ₘ ν)) b
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
π : Measure Bool
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : Bool), ((Kernel.boolKernel μ ν) a) s ∂π = (π {true} • ν + π {false} • μ) s

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
π : Measure Bool
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : Bool), ((Kernel.boolKernel μ ν) a) s ∂π = (π {true} • ν + π {false} • μ) s
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: case pos.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range f ∪ {0})
s : ℕ → SimpleFunc β E := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
f' : ℕ → α → E := fun n ↦ {x | Integrable f (κ x)}.indicator fun x ↦ SimpleFunc.integral (κ x) (s n)
⊢ ∀ (x : α), Tendsto (fun i ↦ f' i x) atTop (𝓝 (∫ (y : β), f y ∂κ x))

Tactic:
]

Goal state after:
case pos.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range f ∪ {0})
s : ℕ → SimpleFunc β E := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
f' : ℕ → α → E := fun n ↦ {x | Integrable f (κ x)}.indicator fun x ↦ SimpleFunc.integral (κ x) (s n)
⊢ ∀ (x : α), Tendsto (fun i ↦ f' i x) atTop (𝓝 (∫ (y : β), f y ∂κ x))
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this✝ : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
f' : SimpleFunc (β × γ) ℝ≥0∞
this : (fun b ↦ ∫⁻ (c : γ), f' (b, c) ∂η (a, b)) = (fun ab ↦ ∫⁻ (c : γ), f' (ab.2, c) ∂η ab) ∘ fun b ↦ (a, b)
⊢ Measurable ((fun ab ↦ ∫⁻ (c : γ), f' (ab.2, c) ∂η ab) ∘ fun b ↦ (a, b))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this✝ : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
f' : SimpleFunc (β × γ) ℝ≥0∞
this : (fun b ↦ ∫⁻ (c : γ), f' (b, c) ∂η (a, b)) = (fun ab ↦ ∫⁻ (c : γ), f' (ab.2, c) ∂η ab) ∘ fun b ↦ (a, b)
⊢ Measurable ((fun ab ↦ ∫⁻ (c : γ), f' (ab.2, c) ∂η ab) ∘ fun b ↦ (a, b))