theorem extracted_formal_statement_85.{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] [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) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    HasFiniteIntegral
      (fun ω => ⨆ I, ∑ i, (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I))
      ((μ.prod μ).prod (Measure.pi fun x => μ.prod μ))) :
  ∀ᵐ (a : (Ω × Ω) × (Fin n → Ω × Ω)) ∂(μ.prod μ).prod (Measure.pi fun x => μ.prod μ),
    ‖⨆ I, ∑ i, (f I (X (a.2 i).1) - f I (X (a.2 i).2)) + (f I (X a.1.1) - f I (X a.1.2) + c I)‖ ≤
      ↑n * (b + b) + (b + b + C) := sorry

State before:
case right
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
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
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ HasFiniteIntegral (fun ω => ⨆ I, ∑ i, (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (f I (X ω.1.1) - f I (X ω.1.2) + c I))
    ((μ.prod μ).prod (Measure.pi fun x => μ.prod μ))

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

State after:
[case right
 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
 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
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ ∀ᵐ (a : (Ω × Ω) × (Fin n → Ω × Ω)) ∂(μ.prod μ).prod (Measure.pi fun x => μ.prod μ),
     ‖⨆ I, ∑ i, (f I (X (a.2 i).1) - f I (X (a.2 i).2)) + (f I (X a.1.1) - f I (X a.1.2) + c I)‖ ≤
       ↑n * (b + b) + (b + b + C)]

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