theorem extracted_formal_statement_214.{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_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω' : Fin n → Ω) (i : ι) (this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1)))
  (w : ∀ (x_1 : Fin n), AEMeasurable (fun a => f i (X (a x_1))) (Measure.pi fun x => μ)) :
  Measurable fun a => ∑ x_1, f i (X (a x_1)) := sorry

State before:
case hg.left.hf.a.hf.h
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_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω' : Fin n → Ω
i : ι
this : ∀ (x : Fin n → Ω), -(↑n * (b + b)) ≤ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (x x_1))
w : ∀ (x_1 : Fin n), AEMeasurable (fun a => f i (X (a x_1))) (Measure.pi fun x => μ)
⊢ Measurable fun a => ∑ x_1, f i (X (a x_1))

Tactic: apply measurable_pi_sum (hf i)

State after:

Relevant lemmas:
measurable_pi_sum : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {ι : Type u_1} [inst_4 : AddCommMonoid Z]
  [inst_5 : MeasurableSpace Z] [inst_6 : MeasurableAdd₂ Z] {X : Ω → Z},
  Measurable X → ∀ (s : Finset ι), Measurable fun ω => ∑ k ∈ s, X (ω k)