theorem extracted_formal_statement_55.{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) (h_original_goal : AEStronglyMeasurable (fun ω => rexp (t' * X ω) * X ω) μ) :
  AEMeasurable (fun ω => rexp (t' * X ω) * X ω) μ := sorry

State before:
case hF_meas.hp
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
⊢ AEStronglyMeasurable (fun ω => rexp (t' * X ω) * X ω) μ

Tactic: rw [aestronglyMeasurable_iff_aemeasurable]

State after:
[case hF_meas.hp
 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 ω => rexp (t' * X ω) * X ω) μ]

Relevant lemmas:
aestronglyMeasurable_iff_aemeasurable : ∀ {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 : BorelSpace β] [inst_4 : SecondCountableTopology β],
  AEStronglyMeasurable f μ ↔ AEMeasurable f μ