theorem extracted_formal_statement_110.{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_3 : Countable ι]
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) (i : ι)
  (h_original_goal :
    StronglyMeasurable fun x => ∫ (x_1 : Fin n → Ω), ∑ k, (f i (X (x k)) - f i (X (x_1 k))) ∂Measure.pi fun x => μ) :
  StronglyMeasurable (Function.uncurry fun x y => ∑ k, (f i (X (y k)) - f i (X (x k)))) := sorry

State before:
case hf.left.hf.hf.hg.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_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
i : ι
⊢ StronglyMeasurable fun x => ∫ (x_1 : Fin n → Ω), ∑ k, (f i (X (x k)) - f i (X (x_1 k))) ∂Measure.pi fun x => μ

Tactic: apply MeasureTheory.StronglyMeasurable.integral_prod_left

State after:
[case hf.left.hf.hf.hg.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_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 i : ι
 ⊢ StronglyMeasurable (Function.uncurry fun x y => ∑ k, (f i (X (y k)) - f i (X (x k))))]

Relevant lemmas:
MeasureTheory.StronglyMeasurable.integral_prod_left : ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3}
  [inst : MeasurableSpace α] [inst_2 : MeasurableSpace β] {μ : Measure α} [inst_4 : NormedAddCommGroup E]
  [inst_5 : NormedSpace ℝ E] [inst_6 : SFinite μ] ⦃f : α → β → E⦄,
  StronglyMeasurable (Function.uncurry f) → StronglyMeasurable fun y => ∫ (x : α), f x y ∂μ