theorem extracted_formal_statement_24.{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 := 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
    ·
  apply Measurable.aemeasurable
  dsimp [Fin.snocEquiv]
  exact measurable_snocEquiv
    ·
  apply Measurable.aestronglyMeasurable
  exact measurable₀ h𝓕 (n + 1) c

State after:

Relevant lemmas:
measurable₀ : ∀ {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} [inst : MeasurableSpace Ω] {X : Ω → Z}
  [inst_4 : Countable ι],
  (∀ (I : ι), Measurable (f I ∘ X)) →
    ∀ (n : ℕ) (c : ι → ℝ), Measurable fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I
Measurable.aestronglyMeasurable : ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m m₀ : MeasurableSpace α}
  {μ : Measure α} {f : α → β} [inst_4 : MeasurableSpace β] [inst_5 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_6 : SecondCountableTopology β] [inst_7 : OpensMeasurableSpace β], Measurable f → AEStronglyMeasurable f μ
measurable_snocEquiv : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {n : ℕ}, Measurable fun f => Fin.snoc f.2 f.1
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
Measurable.aemeasurable : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β}
  {μ : Measure α}, Measurable f → AEMeasurable f μ