theorem extracted_formal_statement_1.{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}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (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) := 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
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (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)

Tactic: rw [← memLp_one_iff_integrable]
    apply MemLp.of_bound (measurable_aux0 hf i k).aestronglyMeasurable b
    filter_upwards with ω
    simp only [norm_eq_abs]
    exact hf' i (X (ω k))

State after:

Relevant lemmas:
MeasureTheory.memLp_one_iff_integrable : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] {f : α → β}, MemLp f 1 μ ↔ Integrable f μ
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|
MeasureTheory.MemLp.of_bound : ∀ {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : Measure α}
  [inst : NormedAddCommGroup E] [inst_2 : IsFiniteMeasure μ] {f : α → E},
  AEStronglyMeasurable f μ → ∀ (C : ℝ), (∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C) → MemLp f p μ
measurable_aux0 : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  {X : Ω → Z}, (∀ (i : ι), Measurable (f i ∘ X)) → ∀ (i : ι) (k : Fin n), Measurable fun ω => f i (X (ω k))