theorem extracted_formal_statement_247.{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))
  (ω : Fin n → Ω) (h_original_goal : Measurable fun ω' => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|‖) :
  Measurable fun a => ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))| := 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)
ω : Fin n → Ω
⊢ Measurable fun ω' => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|‖

Tactic: apply Measurable.norm

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)
 ω : Fin n → Ω
 ⊢ Measurable fun a => ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|]

Relevant lemmas:
Measurable.norm : ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_2 : NormedAddCommGroup α]
  [inst_3 : OpensMeasurableSpace α] [inst_4 : MeasurableSpace β] {f : β → α}, Measurable f → Measurable fun a => ‖f a‖