theorem extracted_formal_statement_107.{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_3 : Countable ι]
  (hf : ∀ (i : ι), Measurable (f i ∘ X))
  (h_original_goal :
    Measurable fun ω => ⨆ i, |∫ (x : Fin n → Ω), ∑ k, (f i (X (ω k)) - f i (X (x k))) ∂Measure.pi fun x => μ|)
  (i : ι) :
  Measurable fun b => |∫ (x : Fin n → Ω), ∑ k, (f i (X (b k)) - f i (X (x k))) ∂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_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
⊢ Measurable fun ω => ⨆ i, |∫ (x : Fin n → Ω), ∑ k, (f i (X (ω k)) - f i (X (x k))) ∂Measure.pi fun x => μ|

Tactic: apply Measurable.iSup

State after:
[case 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_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 ⊢ ∀ (i : ι), Measurable fun b => |∫ (x : Fin n → Ω), ∑ k, (f i (X (b k)) - f i (X (x k))) ∂Measure.pi fun x => μ|]

Relevant lemmas:
Measurable.iSup : ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α}
  [inst_2 : BorelSpace α] {mδ : MeasurableSpace δ} [inst_4 : ConditionallyCompleteLinearOrder α]
  [inst_5 : OrderTopology α] [inst : SecondCountableTopology α] {ι : Sort u_5} [inst : Countable ι] {f : ι → δ → α},
  (∀ (i : ι), Measurable (f i)) → Measurable fun b => ⨆ i, f i b