theorem extracted_formal_statement_289.{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, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ)
      (Measure.pi fun x => μ)) :
  MemLp (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ) 1
    (Measure.pi fun x => μ) := sorry

State before:
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
⊢ Integrable (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ)
    (Measure.pi fun x => μ)

Tactic: simp_rw [← memLp_one_iff_integrable]

State after:
[Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ⊢ MemLp (fun ω => ⨆ i, ∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ) 1
     (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