theorem extracted_formal_statement_78.{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)
  (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)
  (h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t)
  (chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ)
  (h_exp_taylor : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ≤ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ))
  (i : Fin n) (h_single_integ1 : Integrable (fun ω => rexp (τ * X i ω)) μ)
  (h_single_integ2 : Integrable (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h_exp_int : ∫ (ω : Ω), rexp (τ * X i ω) ∂μ ≤ ∫ (ω : Ω), 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ) ∂μ)
  (h_const : Integrable (fun x => 1) μ) (hX_int : Integrable (fun ω => X i ω) μ)
  (h_mul' : Integrable (fun x => τ * X i x) μ) (this : HasFiniteIntegral ((fun x => 1) + fun x => τ * X i x) μ) :
  AEStronglyMeasurable (fun ω => X i ω ^ 2) μ := sorry

State before:
case left
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
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
h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t
chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ
h_exp_taylor : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ≤ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)
i : Fin n
h_single_integ1 : Integrable (fun ω => rexp (τ * X i ω)) μ
h_single_integ2 : Integrable (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h_exp_int : ∫ (ω : Ω), rexp (τ * X i ω) ∂μ ≤ ∫ (ω : Ω), 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ) ∂μ
h_const : Integrable (fun x => 1) μ
hX_int : Integrable (fun ω => X i ω) μ
h_mul' : Integrable (fun x => τ * X i x) μ
this : HasFiniteIntegral ((fun x => 1) + fun x => τ * X i x) μ
⊢ AEStronglyMeasurable (fun ω => X i ω ^ 2) μ

Tactic: exact (Measurable.pow (hXm i) (by measurability)).aestronglyMeasurable

State after:

Relevant lemmas:
Measurable.pow : ∀ {β : Type u_2} {γ : Type u_3} {α : Type u_4} [inst : MeasurableSpace β] [inst_2 : MeasurableSpace γ]
  [inst_3 : Pow β γ] [inst_4 : MeasurablePow β γ] {m : MeasurableSpace α} {f : α → β} {g : α → γ},
  Measurable f → Measurable g → Measurable fun x => f x ^ g x