theorem extracted_formal_statement_4.{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 ω => rexp (τ * S ω)) μ := sorry

State before:
case right
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 ω => rexp (τ * S ω)) μ

Tactic: apply HasFiniteIntegral.mono' (g := fun _ ↦ Real.exp (↑n * τ))
    ·
  simp only [HasFiniteIntegral]
  rw [lintegral_const]
  simp [mul_eq_top, ENNReal.ofReal_lt_top]
    · exact h_sum_bound

State after:

Relevant lemmas:
ENNReal.ofReal_lt_top : ∀ {r : ℝ}, ENNReal.ofReal r < ⊤
MeasureTheory.lintegral_const : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} (c : ℝ≥0∞),
  ∫⁻ (x : α), c ∂μ = c * μ univ
ENNReal.mul_eq_top : ∀ {a b : ℝ≥0∞}, a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0
MeasureTheory.HasFiniteIntegral.mono' : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] {f : α → β} {g : α → ℝ},
  HasFiniteIntegral g μ → (∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ g a) → HasFiniteIntegral f μ