theorem extracted_formal_statement_180.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} [inst_1 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) (σ : Signs n)
  (a : σ ∈ Finset.univ) (h_original_goal : Measurable fun a => ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a k).1)|) (i : ι) :
  Measurable fun b => |∑ k, ↑↑(σ k) * f i (X (b k).1)| := sorry

State before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
σ : Signs n
a✝ : σ ∈ Finset.univ
⊢ Measurable fun a => ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a k).1)|

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
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 ⊢ ∀ (i : ι), Measurable fun b => |∑ k, ↑↑(σ k) * f i (X (b k).1)|]

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_3 : 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