theorem extracted_formal_statement_349.{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)
  (h_original_goal :
    (∫ (x : Fin n → Ω),
        ∫ (y : Fin n → Ω),
          (↑(Fintype.card (Signs n)))⁻¹ *
            ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (y k))| ∂Measure.pi fun x => μ ∂Measure.pi fun x => μ) =
      ∫ (x : Fin n → Ω),
        (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x k))| ∂Measure.pi fun x => μ) :
  (((Measure.pi fun x => μ) Set.univ).toReal •
      ∫ (y : Fin n → Ω),
        (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (y k))| ∂Measure.pi fun x => μ) =
    ∫ (x : Fin n → Ω),
      (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x k))| ∂Measure.pi fun x => μ := sorry

State before:
case hy
Ω : Type u
inst✝³ : 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
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
⊢ (∫ (x : Fin n → Ω),
      ∫ (y : Fin n → Ω),
        (↑(Fintype.card (Signs n)))⁻¹ *
          ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (y k))| ∂Measure.pi fun x => μ ∂Measure.pi fun x => μ) =
    ∫ (x : Fin n → Ω), (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x k))| ∂Measure.pi fun x => μ

Tactic: rw [integral_const]

State after:
[case hy
 Ω : Type u
 inst✝³ : 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
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ⊢ (((Measure.pi fun x => μ) Set.univ).toReal •
       ∫ (y : Fin n → Ω),
         (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (y k))| ∂Measure.pi fun x => μ) =
     ∫ (x : Fin n → Ω), (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (x k))| ∂Measure.pi fun x => μ]

Relevant lemmas:
MeasureTheory.integral_const : ∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [hE : CompleteSpace E]
  [inst_4 : NormedSpace ℝ E] {m : MeasurableSpace α} {μ : Measure α} (c : E), ∫ (x : α), c ∂μ = (μ Set.univ).toReal • c