theorem extracted_formal_statement_163.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n_1 : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) (n : ℕ)
  (ih :
    ∀ (c : ι → ℝ),
      (∃ C, ∀ (I : ι), |c I| ≤ C) →
        (∫ (x : Fin n → Ω × Ω),
            (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
          ∫ (x : Fin n → Ω × Ω),
            (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
              x ∂Measure.pi fun x => μ.prod μ)
  (c : ι → ℝ) (C : ℝ) (hC : ∀ (I : ι), |c I| ≤ C) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  ∫ (x : Fin (n + 1) → Ω × Ω),
        (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
          x ∂Measure.map (⇑(Fin.snocEquiv fun x => Ω × Ω)) ((μ.prod μ).prod μ2n) =
      ∫ (x : (Ω × Ω) × (Fin n → Ω × Ω)),
        ((fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) ∘ ⇑(Fin.snocEquiv fun x => Ω × Ω))
          x ∂(μ.prod μ).prod μ2n →
    AEStronglyMeasurable (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
      (Measure.map (⇑(Fin.snocEquiv fun x => Ω × Ω)) ((μ.prod μ).prod μ2n)) := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n✝ : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
ih :
  ∀ (c : ι → ℝ),
    (∃ C, ∀ (I : ι), |c I| ≤ C) →
      (∫ (x : Fin n → Ω × Ω),
          (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
        ∫ (x : Fin n → Ω × Ω),
          (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
            x ∂Measure.pi fun x => μ.prod μ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
⊢ ∫ (x : Fin (n + 1) → Ω × Ω),
      (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
        x ∂Measure.map (⇑(Fin.snocEquiv fun x => Ω × Ω)) ((μ.prod μ).prod μ2n) =
    ∫ (x : (Ω × Ω) × (Fin n → Ω × Ω)),
      ((fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) ∘ ⇑(Fin.snocEquiv fun x => Ω × Ω))
        x ∂(μ.prod μ).prod μ2n

Tactic: apply integral_map

State after:
[case hφ
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 n✝ : ℕ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 ih :
   ∀ (c : ι → ℝ),
     (∃ C, ∀ (I : ι), |c I| ≤ C) →
       (∫ (x : Fin n → Ω × Ω),
           (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
         ∫ (x : Fin n → Ω × Ω),
           (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
             x ∂Measure.pi fun x => μ.prod μ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
 ⊢ AEMeasurable (⇑(Fin.snocEquiv fun x => Ω × Ω)) ((μ.prod μ).prod μ2n),
 case hfm
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 n✝ : ℕ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 ih :
   ∀ (c : ι → ℝ),
     (∃ C, ∀ (I : ι), |c I| ≤ C) →
       (∫ (x : Fin n → Ω × Ω),
           (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) x ∂Measure.pi fun x => μ.prod μ) =
         ∫ (x : Fin n → Ω × Ω),
           (fun ω => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
             x ∂Measure.pi fun x => μ.prod μ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
 ⊢ AEStronglyMeasurable (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I)
     (Measure.map (⇑(Fin.snocEquiv fun x => Ω × Ω)) ((μ.prod μ).prod μ2n))]

Relevant lemmas:
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