theorem extracted_formal_statement_25.{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)) → ‖rexp (↑n * τ)‖ₑ * μ univ < ⊤ := 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)
⊢ ‖rexp (↑n * τ)‖ₑ * μ univ < ⊤

Tactic: simp [mul_eq_top, ENNReal.ofReal_lt_top]

State after:

Relevant lemmas:
ENNReal.ofReal_lt_top : ∀ {r : ℝ}, ENNReal.ofReal r < ⊤
ENNReal.mul_eq_top : ∀ {a b : ℝ≥0∞}, a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0