theorem extracted_formal_statement_212.{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)))
  (x_1 : Fin n) : AEMeasurable (fun a => f i (X (a x_1))) (Measure.pi fun x => μ) := sorry

State before:
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))
x_1 : Fin n
⊢ AEMeasurable (fun a => f i (X (a x_1))) (Measure.pi fun x => μ)

Tactic: exact Integrable.aemeasurable (integrable_aux0 hf hf' i x_1)

State after:

Relevant lemmas:
MeasureTheory.Integrable.aemeasurable : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] [inst_4 : MeasurableSpace β] [inst_5 : BorelSpace β] {f : α → β},
  Integrable f μ → AEMeasurable f μ
integrable_aux0 : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  {μ : Measure Ω} [inst_4 : IsProbabilityMeasure μ] {X : Ω → Z},
  (∀ (i : ι), Measurable (f i ∘ X)) →
    ∀ {b : ℝ},
      (∀ (i : ι) (z : Z), |f i z| ≤ b) →
        ∀ (i : ι) (k : Fin n), Integrable (fun ω => f i (X (ω k))) (Measure.pi fun x => μ)