theorem extracted_formal_statement_52.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} [inst_1 : Countable ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) (n : ℕ)
  (c : ι → ℝ) (h_original_goal : Measurable fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I) (i : ι) :
  Measurable fun b => ∑ i_1, (f i (X (b i_1).1) - f i (X (b i_1).2)) + c i := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝¹ : MeasurableSpace Ω
X : Ω → Z
inst✝ : Countable ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
n : ℕ
c : ι → ℝ
⊢ Measurable fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).2)) + c I

Tactic: apply Measurable.iSup

State after:
[case hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 X : Ω → Z
 inst✝ : Countable ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 n : ℕ
 c : ι → ℝ
 ⊢ ∀ (i : ι), Measurable fun b => ∑ i_1, (f i (X (b i_1).1) - f i (X (b i_1).2)) + c i]

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