theorem extracted_formal_statement_76.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : b ≥ 0)
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (hn : ¬n = 0)
  (hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C)
  (hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y)
  (hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k))))
  (hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k))))
  (h_original_goal :
    (∫ (x : Fin n → Ω),
        ∫ (x_1 : Fin n → Ω),
          ⨆ i, |∑ k, (f i (X (x k)) - f i (X (x_1 k)))| ∂Measure.pi fun x => μ ∂Measure.pi fun x => μ) =
      ∫ (x : Fin n → Ω × Ω), ⨆ i, |∑ k, (f i (X (x k).1) - f i (X (x k).2))| ∂Measure.pi fun x => μ.prod μ) :
  Integrable (fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|)
    ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ)) := sorry

State before:
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : b ≥ 0
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
hn : ¬n = 0
hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
⊢ (∫ (x : Fin n → Ω),
      ∫ (x_1 : Fin n → Ω),
        ⨆ i, |∑ k, (f i (X (x k)) - f i (X (x_1 k)))| ∂Measure.pi fun x => μ ∂Measure.pi fun x => μ) =
    ∫ (x : Fin n → Ω × Ω), ⨆ i, |∑ k, (f i (X (x k).1) - f i (X (x k).2))| ∂Measure.pi fun x => μ.prod μ

Tactic: rw [←
  integral_prod fun ω : (Fin n → Ω) × (Fin n → Ω) ↦ ⨆ i, |∑ k : Fin n, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|]
    let mp := measurePreserving_arrowProdEquivProdArrow Ω Ω (Fin n) (fun _ ↦ μ) (fun _ ↦ μ)
    rw [← mp.map_eq, integral_map mp.measurable.aemeasurable]

State after:
[Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : b ≥ 0
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 hn : ¬n = 0
 hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
 hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
 hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
 hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
 mp : MeasurePreserving (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ)
   ((Measure.pi fun i => μ).prod (Measure.pi fun i => μ)) :=
   measurePreserving_arrowProdEquivProdArrow Ω Ω (Fin n) (fun x => μ) fun x => μ
 ⊢ (∫ (x : Fin n → Ω × Ω),
       ⨆ i,
         |∑ k,
             (f i (X (((MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n)) x).1 k)) -
               f i
                 (X (((MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n)) x).2 k)))| ∂Measure.pi fun i => μ.prod μ) =
     ∫ (x : Fin n → Ω × Ω), ⨆ i, |∑ k, (f i (X (x k).1) - f i (X (x k).2))| ∂Measure.pi fun x => μ.prod μ,
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : b ≥ 0
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 hn : ¬n = 0
 hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
 hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
 hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
 hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
 mp : MeasurePreserving (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ)
   ((Measure.pi fun i => μ).prod (Measure.pi fun i => μ)) :=
   measurePreserving_arrowProdEquivProdArrow Ω Ω (Fin n) (fun x => μ) fun x => μ
 ⊢ AEStronglyMeasurable (fun z => ⨆ i, |∑ k, (f i (X (z.1 k)) - f i (X (z.2 k)))|)
     (Measure.map (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ)),
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : b ≥ 0
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 hn : ¬n = 0
 hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
 hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
 hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
 hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
 ⊢ Integrable (fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|)
     ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))]

Relevant lemmas:
MeasureTheory.integral_prod : ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α]
  [inst_4 : MeasurableSpace β] {μ : Measure α} {ν : Measure β} [inst_5 : NormedAddCommGroup E] [inst_6 : SFinite ν]
  [inst_7 : NormedSpace ℝ E] [inst_8 : SFinite μ] (f : α × β → E),
  Integrable f (μ.prod ν) → ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ
MeasureTheory.measurePreserving_arrowProdEquivProdArrow : ∀ (α : Type u_4) (β : Type u_5) (γ : Type u_6)
  [inst : MeasurableSpace α] [inst_4 : MeasurableSpace β] [inst_5 : Fintype γ] (μ : γ → Measure α) (ν : γ → Measure β)
  [inst_6 : ∀ (i : γ), SigmaFinite (μ i)] [inst_7 : ∀ (i : γ), SigmaFinite (ν i)],
  MeasurePreserving (⇑(MeasurableEquiv.arrowProdEquivProdArrow α β γ)) (Measure.pi fun i => (μ i).prod (ν i))
    ((Measure.pi fun i => μ i).prod (Measure.pi fun i => ν i))
MeasureTheory.integral_map : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_4 : NormedSpace ℝ G]
  {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [inst_5 : MeasurableSpace β] {φ : α → β},
  AEMeasurable φ μ →
    ∀ {f : β → G}, AEStronglyMeasurable f (Measure.map φ μ) → ∫ (y : β), f y ∂Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ