theorem extracted_formal_statement_360.{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)
  (i : Signs n) (hi : i ∈ Finset.univ) (i_1 : ι) (k : Fin n)
  (h_original_goal : Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))) : Measurable fun a => f i_1 (X (a.2 k)) := 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
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
i : Signs n
hi : i ∈ Finset.univ
i_1 : ι
k : Fin n
⊢ Measurable fun b => ↑↑(i k) * f i_1 (X (b.2 k))

Tactic: apply Measurable.mul

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
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 i : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => ↑↑(i k),
 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
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 i : Signs n
 hi : i ∈ Finset.univ
 i_1 : ι
 k : Fin n
 ⊢ Measurable fun a => f i_1 (X (a.2 k))]

Relevant lemmas:
Measurable.mul : ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_4 : Mul M] {m : MeasurableSpace α}
  {f g : α → M} [inst_5 : MeasurableMul₂ M], Measurable f → Measurable g → Measurable fun a => f a * g a