theorem extracted_formal_statement_5.{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 : AEStronglyMeasurable (fun ω => rexp (t * X ω) * X ω) μ) :
  AEMeasurable (fun x => t * X x) μ := sorry

State before:
case hF'_meas
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
⊢ AEStronglyMeasurable (fun ω => rexp (t * X ω) * X ω) μ

Tactic: apply AEMeasurable.aestronglyMeasurable
    apply AEMeasurable.mul _ hX
    apply Measurable.comp_aemeasurable' _ _

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
 ⊢ Measurable rexp,
 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 x => t * X x) μ]

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
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
AEMeasurable.aestronglyMeasurable : ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m₀ : MeasurableSpace α}
  {μ : Measure α} {f : α → β} [inst_1 : MeasurableSpace β] [inst_2 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_3 : OpensMeasurableSpace β] [inst_4 : SecondCountableTopology β], AEMeasurable f μ → AEStronglyMeasurable f μ