-- Mathlib/MeasureTheory/Measure/Prod.lean
theorem lintegral_prod_of_measurable :
    ∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ z, f z ∂μ.prod ν = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ

Proof state:
case refine_1
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m :
  ∀ {α : Type ?u.165217} {β : Type ?u.165216} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},
    Measurable (Prod.mk x)
⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)⦄,
    MeasurableSet s →
      (fun f => ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ) (s.indicator fun x => c)

Tactic:
intro c s hs

Result:
case refine_1
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m :
  ∀ {α : Type ?u.165217} {β : Type ?u.165216} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},
    Measurable (Prod.mk x)
c : ℝ≥0∞
s : Set (α × β)
hs : MeasurableSet s
⊢ ∫⁻ (z : α × β), s.indicator (fun x => c) z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), s.indicator (fun x => c) (x, y) ∂ν ∂μ

Proof state:
case refine_3
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
⊢ ∀ ⦃f : ℕ → α × β → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) →
      Monotone f →
        (∀ (n : ℕ), (fun f => ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ) (f n)) →
          (fun f => ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ) fun x => ⨆ n, f n x

Tactic:
intro f hf h2f h3f

Result:
case refine_3
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f : ℕ → α × β → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h2f : Monotone f
h3f : ∀ (n : ℕ), (fun f => ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ) (f n)
⊢ ∫⁻ (z : α × β), (fun x => ⨆ n, f n x) z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), (fun x => ⨆ n, f n x) (x, y) ∂ν ∂μ

Proof state:
case refine_3
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f : ℕ → α × β → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h2f : Monotone f
h3f : ∀ (n : ℕ), (fun f => ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ) (f n)
⊢ ∫⁻ (z : α × β), (fun x => ⨆ n, f n x) z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), (fun x => ⨆ n, f n x) (x, y) ∂ν ∂μ

Tactic:
have kf : ∀ x n, Measurable fun y => f n (x, y) := fun x n => (hf n).comp m

Result:
case refine_3
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f : ℕ → α × β → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h2f : Monotone f
h3f : ∀ (n : ℕ), (fun f => ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ) (f n)
kf : ∀ (x : α) (n : ℕ), Measurable fun y => f n (x, y)
⊢ ∫⁻ (z : α × β), (fun x => ⨆ n, f n x) z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), (fun x => ⨆ n, f n x) (x, y) ∂ν ∂μ

Proof state:
case refine_2
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f g : α × β → ℝ≥0∞
hf : Measurable f
a✝ : Measurable g
h2f : ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ
h2g : ∫⁻ (z : α × β), g z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), g (x, y) ∂ν ∂μ
⊢ ∫⁻ (z : α × β), (f + g) z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), (f + g) (x, y) ∂ν ∂μ

Tactic:
simp only [Pi.add_apply]

Result:
case refine_2
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f g : α × β → ℝ≥0∞
hf : Measurable f
a✝ : Measurable g
h2f : ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ
h2g : ∫⁻ (z : α × β), g z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), g (x, y) ∂ν ∂μ
⊢ ∫⁻ (z : α × β), f z + g z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) + g (x, y) ∂ν ∂μ

Proof state:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
⊢ ∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ

Tactic:
have m := @measurable_prod_mk_left

Result:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m :
  ∀ {α : Type ?u.165217} {β : Type ?u.165216} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},
    Measurable (Prod.mk x)
⊢ ∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ