theorem extracted_formal_statement_314.{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} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : Fin n → Ω) : Integrable (fun a => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|‖) (Measure.pi fun x => μ) := sorry

State before:
case refine_2.h.hf
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω
⊢ Integrable (fun a => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|‖) (Measure.pi fun x => μ)

Tactic: exact integrable_aux5 hf hf' ω

State after:

Relevant lemmas:
integrable_aux5 : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  {μ : Measure Ω} [inst_4 : IsProbabilityMeasure μ] [inst_5 : Nonempty ι] [inst_6 : Countable ι] {X : Ω → Z},
  (∀ (i : ι), Measurable (f i ∘ X)) →
    ∀ {b : ℝ},
      (∀ (i : ι) (z : Z), |f i z| ≤ b) →
        ∀ (ω : Fin n → Ω), Integrable (fun ω' => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|‖) (Measure.pi fun x => μ)