theorem extracted_formal_statement_32.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (h_original_goal : AEMeasurable (fun ω => rexp (t * X ω) * X ω) μ) : AEMeasurable (fun a => rexp (t * X a)) μ := sorry

State before:
case hF'_meas.hf
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
⊢ AEMeasurable (fun ω => rexp (t * X ω) * X ω) μ

Tactic: apply AEMeasurable.mul _ hX

State after:
[Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 t a b : ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 c : ℝ := ‖a‖ ⊔ ‖b‖
 e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
 ⊢ AEMeasurable (fun a => rexp (t * X a)) μ]

Relevant lemmas:
AEMeasurable.mul : ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_1 : Mul M] {m : MeasurableSpace α}
  {f g : α → M} {μ : Measure α} [inst_2 : MeasurableMul₂ M],
  AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (fun a => f a * g a) μ