theorem extracted_formal_statement_181.{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) (i : ι) (h_original_goal : Measurable fun b => |∑ k, ↑↑(σ k) * f i (X (b k).1)|) :
  Measurable fun x => ∑ k, ↑↑(σ k) * f i (X (x k).1) := sorry

State before:
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)|

Tactic: apply Measurable.abs

State after:
[case hf.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 x => ∑ k, ↑↑(σ k) * f i (X (x k).1)]

Relevant lemmas:
Measurable.abs : ∀ {α : Type u_1} {β : Type u_2} [inst : Lattice α] [inst_2 : AddGroup α] [inst_3 : MeasurableSpace α]
  [inst_4 : MeasurableSpace β] {f : β → α} [inst_5 : MeasurableNeg α] [inst_6 : MeasurableSup₂ α],
  Measurable f → Measurable fun x => |f x|