-- Mathlib/Probability/Kernel/MeasurableIntegral.lean
theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
n : ℕ
this : ∀ (x : α), Finset.filter (fun x => x ≠ 0) (s' n x).range ⊆ (s n).range
x : E
x✝ : x ∈ (s n).range
⊢ Measurable fun x_1 => (κ x_1) (Prod.mk x_1 ⁻¹' (⇑(s n) ⁻¹' {x}))

Tactic:
apply Kernel.measurable_kernel_prod_mk_left

Result:
case ht
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
n : ℕ
this : ∀ (x : α), Finset.filter (fun x => x ≠ 0) (s' n x).range ⊆ (s n).range
x : E
x✝ : x ∈ (s n).range
⊢ MeasurableSet (⇑(s n) ⁻¹' {x})

Proof state:
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✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
⊢ StronglyMeasurable fun x => ∫ (y : β), f x y ∂κ x

Tactic:
let f' : ℕ → α → E := fun n =>
  {x | Integrable (f x) (κ x)}.indicator fun x => (s' n x).integral (κ x)

Result:
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✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
⊢ StronglyMeasurable fun x => ∫ (y : β), f x y ∂κ x

Proof state:
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✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : α
hfx : Integrable (f x) (κ x)
this : ∀ (n : ℕ), Integrable (⇑(s' n x)) (κ x)
y : β
⊢ uncurry f (x, y) ∈ closure (range (uncurry f) ∪ {0})

Tactic:
apply subset_closure

Result:
case pos.refine_2.a
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : α
hfx : Integrable (f x) (κ x)
this : ∀ (n : ℕ), Integrable (⇑(s' n x)) (κ x)
y : β
⊢ uncurry f (x, y) ∈ range (uncurry f) ∪ {0}

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
n : ℕ
x : α
⊢ Finset.filter (fun x => x ≠ 0) (s' n x).range ⊆ (s n).range

Tactic:
refine Finset.Subset.trans (Finset.filter_subset _ _) ?_

Result:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) (κ x)}.indicator fun x => SimpleFunc.integral (κ x) (s' n x)
n : ℕ
x : α
⊢ (s' n x).range ⊆ (s n).range