theorem extracted_formal_statement_302.{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 : Measurable fun p => ⨆ i, |∑ k, (f i (X (p.2 k)) - f i (X (p.1 k)))|) (i : ι) :
  Measurable fun b => |∑ k, (f i (X (b.2 k)) - f i (X (b.1 k)))| := sorry

State before:
case refine_1.hf.hf.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_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
⊢ Measurable fun p => ⨆ i, |∑ k, (f i (X (p.2 k)) - f i (X (p.1 k)))|

Tactic: apply Measurable.iSup

State after:
[case refine_1.hf.hf.hf.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_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
 ⊢ ∀ (i : ι), Measurable fun b => |∑ k, (f i (X (b.2 k)) - f i (X (b.1 k)))|]

Relevant lemmas:
Measurable.iSup : ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α}
  [inst_4 : BorelSpace α] {mδ : MeasurableSpace δ} [inst_5 : ConditionallyCompleteLinearOrder α]
  [inst_6 : 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