theorem extracted_formal_statement_16.{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] [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}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    StronglyMeasurable fun a =>
      ∫ (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)) + (f I (X a.1) - f I (X a.2) + c I))
          x ∂Measure.pi fun x => μ.prod μ) :
  Measurable fun x =>
    ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (f I (X x.1.1) - f I (X x.1.2) + c I) := sorry

State before:
case left.hf
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
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
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ StronglyMeasurable fun a =>
    ∫ (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)) + (f I (X a.1) - f I (X a.2) + c I))
        x ∂Measure.pi fun x => μ.prod μ

Tactic: apply
  @StronglyMeasurable.integral_prod_right' (Ω × Ω) (Fin n → (Ω × Ω)) ℝ _ _ (Measure.pi fun _ ↦ μ.prod μ) _ _ _
    (fun ω ↦
      (2⁻¹ ^ n *
        ∑ σ : Fin n → ({-1, 1} : Finset ℤ),
          ⨆ I, ∑ i : Fin n, ↑↑(σ i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I)))
    apply Measurable.stronglyMeasurable
    apply Measurable.const_mul

State after:
[case left.hf.hf.hf
 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
 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
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ Measurable fun x =>
     ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (x.2 i).1) - f I (X (x.2 i).2)) + (f I (X x.1.1) - f I (X x.1.2) + c I)]

Relevant lemmas:
Measurable.stronglyMeasurable : ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α}
  [inst : MeasurableSpace β] [inst_4 : TopologicalSpace β] [inst_5 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_6 : SecondCountableTopology β] [inst_7 : OpensMeasurableSpace β], Measurable f → StronglyMeasurable f
Measurable.const_mul : ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_4 : Mul M]
  {m : MeasurableSpace α} {f : α → M} [inst_5 : MeasurableMul M], Measurable f → ∀ (c : M), Measurable fun x => c * f x
MeasureTheory.StronglyMeasurable.integral_prod_right' : ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3}
  [inst : MeasurableSpace α] [inst_4 : MeasurableSpace β] {ν : Measure β} [inst_5 : NormedAddCommGroup E]
  [inst_6 : NormedSpace ℝ E] [inst_7 : SFinite ν] ⦃f : α × β → E⦄,
  StronglyMeasurable f → StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν