theorem extracted_formal_statement_320.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {X : Ω → Z} [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X))
  (h_original_goal : Measurable fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|) (i : ι) :
  Measurable fun b => |∑ k, (f i (X (b.1 k)) - f i (X (b.2 k)))| := sorry

State before:
Ω : Type u
inst✝ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
X : Ω → Z
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
⊢ Measurable fun ω => ⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|

Tactic: apply Measurable.iSup

State after:
[case hf
 Ω : Type u
 inst✝ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 X : Ω → Z
 inst_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 ⊢ ∀ (i : ι), Measurable fun b => |∑ k, (f i (X (b.1 k)) - f i (X (b.2 k)))|]

Relevant lemmas:
Measurable.iSup : ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α}
  [inst_1 : BorelSpace α] {mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α]
  [inst_4 : 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