theorem extracted_formal_statement_325.{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)
  (h_original_goal :
    Integrable (fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|)
      ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))) :
  MemLp (fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|) 1
    ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ)) := sorry

State before:
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
⊢ Integrable (fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|)
    ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))

Tactic: rw [← memLp_one_iff_integrable]

State after:
[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
 ⊢ MemLp (fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|) 1
     ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))]

Relevant lemmas:
MeasureTheory.memLp_one_iff_integrable : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] {f : α → β}, MemLp f 1 μ ↔ Integrable f μ