theorem extracted_formal_statement_82.{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)))) :
  let mp := measurePreserving_arrowProdEquivProdArrow Ω Ω (Fin n) (fun x => μ) fun x => μ;
  AEMeasurable
      (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))|)
      (Measure.map (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ)) →
    Measurable fun a => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))| := sorry

State before:
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)))
mp : MeasurePreserving (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ)
  ((Measure.pi fun i => μ).prod (Measure.pi fun i => μ)) :=
  measurePreserving_arrowProdEquivProdArrow Ω Ω (Fin n) (fun x => μ) fun x => μ
⊢ AEMeasurable
    (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))|)
    (Measure.map (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ))

Tactic: apply Measurable.aemeasurable
    apply Measurable.add
    ·
  apply Measurable.const_mul
  exact measurable_aux20 hf

State after:
[case hf.h.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)))
 mp : MeasurePreserving (⇑(MeasurableEquiv.arrowProdEquivProdArrow Ω Ω (Fin n))) (Measure.pi fun i => μ.prod μ)
   ((Measure.pi fun i => μ).prod (Measure.pi fun i => μ)) :=
   measurePreserving_arrowProdEquivProdArrow Ω Ω (Fin n) (fun x => μ) fun x => μ
 ⊢ Measurable fun a => (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (a.2 k))|]

Relevant lemmas:
Measurable.add : ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_4 : Add M] {m : MeasurableSpace α}
  {f g : α → M} [inst_5 : MeasurableAdd₂ M], Measurable f → Measurable g → Measurable fun a => f a + g a
Measurable.aemeasurable : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β}
  {μ : Measure α}, Measurable f → AEMeasurable f μ
measurable_aux20 : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  [inst_4 : Countable ι] {X : Ω → Z},
  (∀ (i : ι), Measurable (f i ∘ X)) → Measurable fun ω => ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|
Measurable.const_mul : ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_4 : Mul M]
  {m : MeasurableSpace α} {f : α → M} [inst_5 : MeasurableMul M], Measurable f → ∀ (c : M), Measurable fun x => c * f x