theorem extracted_formal_statement_57.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (t' : ℝ) : AEMeasurable (fun a => rexp (t' * X a)) μ := sorry

State before:
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
t' : ℝ
⊢ AEMeasurable (fun a => rexp (t' * X a)) μ

Tactic: exact Measurable.comp_aemeasurable' measurable_exp (hX.const_mul t')

State after:

Relevant lemmas:
Measurable.comp_aemeasurable' : ∀ {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α}
  [inst : MeasurableSpace β] {μ : Measure α} [inst_1 : MeasurableSpace δ] {f : α → δ} {g : δ → β},
  Measurable g → AEMeasurable f μ → AEMeasurable (fun x => g (f x)) μ
Real.measurable_exp : Measurable rexp