theorem extracted_formal_statement_155.{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 : ℕ} (ω : Ω × Ω)
  [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) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  Integrable
    (fun a =>
      2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
    μ2n := sorry

State before:
case h.h.hg
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
ω : Ω × Ω
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
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
⊢ Integrable
    (fun a =>
      2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
    μ2n

Tactic: apply integrable₁ ω h𝓕 h𝓕' hC

State after:

Relevant lemmas:
integrable₁ : ∀ {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u} [inst : MeasurableSpace Ω] {X : Ω → Z}
  {μ : Measure Ω} [inst_4 : IsProbabilityMeasure μ] {n : ℕ} (ω : Ω × Ω) [inst_5 : Countable ι] [inst_6 : Nonempty ι],
  (∀ (I : ι), Measurable (f I ∘ X)) →
    ∀ {b : ℝ},
      (∀ (I : ι) (z : Z), |f I z| ≤ b) →
        ∀ {c : ι → ℝ} {C : ℝ},
          (∀ (I : ι), |c I| ≤ C) →
            let μ2n := Measure.pi fun x => μ.prod μ;
            Integrable
              (fun a =>
                2⁻¹ ^ n *
                  ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
              μ2n