theorem extracted_formal_statement_41.{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 : Integrable (fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω|) μ) :
  AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ := sorry

State before:
case bound_integrable
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
⊢ Integrable (fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω|) μ

Tactic: apply MeasureTheory.Integrable.bdd_mul'

State after:
[case bound_integrable.hg
 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
 ⊢ Integrable (fun x => |X x|) μ,
 case bound_integrable.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
 ⊢ AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ,
 case bound_integrable.hf_bound
 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
 ⊢ ∀ᵐ (x : Ω) ∂μ, ‖rexp ((2 * |t| + 1) * |X x|)‖ ≤ ?bound_integrable.c,
 case bound_integrable.c
 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
 ⊢ ℝ]

Relevant lemmas:
MeasureTheory.Integrable.bdd_mul' : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {𝕜 : Type u_6}
  [inst : NormedRing 𝕜] {f g : α → 𝕜} {c : ℝ},
  Integrable g μ → AEStronglyMeasurable f μ → (∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c) → Integrable (fun x => f x * g x) μ