theorem extracted_formal_statement_204.{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 :
    HasFiniteIntegral (fun a => ⨆ i, |∫ (x : Fin n → Ω), ∑ k, (f i (X (a k)) - f i (X (x k))) ∂Measure.pi fun x => μ|)
      (Measure.pi fun x => μ)) :
  ∀ᵐ (a : Fin n → Ω) ∂Measure.pi fun x => μ,
    ‖⨆ i, |∫ (x : Fin n → Ω), ∑ k, (f i (X (a k)) - f i (X (x k))) ∂Measure.pi fun x => μ|‖ ≤ ↑n * (b + b) := sorry

State before:
case right
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
⊢ HasFiniteIntegral (fun a => ⨆ i, |∫ (x : Fin n → Ω), ∑ k, (f i (X (a k)) - f i (X (x k))) ∂Measure.pi fun x => μ|)
    (Measure.pi fun x => μ)

Tactic: apply @MeasureTheory.hasFiniteIntegral_of_bounded _ _ _ _ _ _ _ (n * (b + b))

State after:
[case right
 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
 ⊢ ∀ᵐ (a : Fin n → Ω) ∂Measure.pi fun x => μ,
     ‖⨆ i, |∫ (x : Fin n → Ω), ∑ k, (f i (X (a k)) - f i (X (x k))) ∂Measure.pi fun x => μ|‖ ≤ ↑n * (b + b)]

Relevant lemmas:
MeasureTheory.hasFiniteIntegral_of_bounded : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] [inst_4 : IsFiniteMeasure μ] {f : α → β} {C : ℝ},
  (∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ C) → HasFiniteIntegral f μ