theorem extracted_formal_statement_23.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) →
          HasFiniteIntegral (fun x => rexp (↑n * τ)) μ → ∫⁻ (a : Ω), ‖rexp (↑n * τ)‖ₑ ∂μ < ⊤ := sorry

State before:
case right.hg
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ HasFiniteIntegral (fun x => rexp (↑n * τ)) μ

Tactic: simp only [HasFiniteIntegral]

State after:
[case right.hg
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ ∫⁻ (a : Ω), ‖rexp (↑n * τ)‖ₑ ∂μ < ⊤]

Relevant lemmas: