theorem extracted_formal_statement_493.{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} (hb : b ≥ 0)
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (hn : ¬n = 0)
  (hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C)
  (hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y)
  (hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k))))
  (hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k))))
  (h_original_goal :
    ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω =>
            ((↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) +
              (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) =
      ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
          (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)
            x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) +
        ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
          (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
            x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ)) :
  Integrable (fun a => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.1 k))|)
    ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ)) := sorry

State before:
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
hb : b ≥ 0
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
hn : ¬n = 0
hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
⊢ ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
      (fun ω =>
          ((↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) +
            (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
        x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) =
    ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ) +
      ∫ (x : (Fin n → Ω) × (Fin n → Ω)),
        (fun ω => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.2 k))|)
          x ∂(Measure.pi fun x => μ).prod (Measure.pi fun x => μ)

Tactic: apply integral_add

State after:
[case hf
 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
 hb : b ≥ 0
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 hn : ¬n = 0
 hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
 hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
 hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
 hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
 ⊢ Integrable (fun a => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.1 k))|)
     ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ)),
 case hg
 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
 hb : b ≥ 0
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 hn : ¬n = 0
 hr : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), y ≤ C
 hr' : ∃ C, ∀ (ω : Fin n → Ω) (k : Fin n), ∀ y ∈ Set.range fun i => f i (X (ω k)), C ≤ y
 hq : ∀ (ω : Fin n → Ω) (k : Fin n), BddAbove (Set.range fun i => f i (X (ω k)))
 hq' : ∀ (ω : Fin n → Ω) (k : Fin n), BddBelow (Set.range fun i => f i (X (ω k)))
 ⊢ Integrable (fun a => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|)
     ((Measure.pi fun x => μ).prod (Measure.pi fun x => μ))]

Relevant lemmas:
MeasureTheory.integral_add : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_4 : NormedSpace ℝ G]
  {m : MeasurableSpace α} {μ : Measure α} {f g : α → G},
  Integrable f μ → Integrable g μ → ∫ (a : α), f a + g a ∂μ = ∫ (a : α), f a ∂μ + ∫ (a : α), g a ∂μ