theorem extracted_formal_statement_58.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (ε τ : ℝ) (hτ : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  (Integrable (fun ω => rexp (τ * S ω)) μ → (μ {ω | ε ≤ S ω}).toReal ≤ rexp (-τ * ε) * mgf S μ τ) →
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ := sorry

State before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
σ ε τ : ℝ
hτ : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
ht_sq : 0 ≤ σ ^ 2
chernoff_bound : Integrable (fun ω => rexp (τ * S ω)) μ → (μ {ω | ε ≤ S ω}).toReal ≤ rexp (-τ * ε) * mgf S μ τ
⊢ Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ

Tactic: exact chernoff_bound

State after:

Relevant lemmas: