-- File: FoML/FoML/Rademacher.lean
Theorem statement:
theorem extracted_formal_statement_337.{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) :
  (∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω' k)) ∂Measure.pi fun x => μ) =
    ∑ k, ∫ (ω' : Fin n → Ω), f i (X (ω' k)) ∂Measure.pi fun x => μ := sorry

Tactic 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
ω : Fin n → Ω
i : ι
⊢ (∫ (ω' : Fin n → Ω), ∑ k, f i (X (ω' k)) ∂Measure.pi fun x => μ) =
    ∑ k, ∫ (ω' : Fin n → Ω), f i (X (ω' k)) ∂Measure.pi fun x => μ

Tactic: exact integral_finset_sum Finset.univ (fun i_1 a ↦ integrable_aux0 hf hf' i i_1)

Tactic state after:

Premises used:
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_2 : 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)
MeasureTheory.integral_finset_sum : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G]
  [inst_2 : NormedSpace ℝ G] {m : MeasurableSpace α} {μ : Measure α} {ι : Type u_6} (s : Finset ι) {f : ι → α → G},
  (∀ i ∈ s, Integrable (f i) μ) → ∫ (a : α), ∑ i ∈ s, f i a ∂μ = ∑ i ∈ s, ∫ (a : α), f i a ∂μ