theorem extracted_formal_statement_9.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (chernoff_bound :
    τ ≥ 0 →
      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)
  (h_chernoff :
    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)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ) (t : ℝ) (hXmeas : ∀ (i : Fin n), Measurable (X i)) :
  mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t := sorry

State before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
chernoff_bound :
  τ ≥ 0 →
    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
h_chernoff :
  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
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
t : ℝ
hXmeas : ∀ (i : Fin n), Measurable (X i)
⊢ mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t

Tactic: rw [← iIndepFun.mgf_sum hXi hXmeas Finset.univ]
    congr with x
    rw [Finset.sum_apply]

State after:

Relevant lemmas:
ProbabilityTheory.iIndepFun.mgf_sum : ∀ {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} {μ : Measure Ω} {t : ℝ}
  {X : ι → Ω → ℝ},
  iIndepFun X μ → (∀ (i : ι), Measurable (X i)) → ∀ (s : Finset ι), mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t
Finset.sum_apply : ∀ {α : Type u_4} {β : α → Type u_5} {γ : Type u_6} [inst : (a : α) → AddCommMonoid (β a)] (a : α)
  (s : Finset γ) (g : γ → (a : α) → β a), (∑ c ∈ s, g c) a = ∑ c ∈ s, g c a