theorem extracted_formal_statement_53.{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))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1) (h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0)
  (h_box : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ∈ Icc 0 1) : HasFiniteIntegral (fun ω => X i ω ^ 2) μ := sorry

State before:
case right
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)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1
h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0
h_box : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ∈ Icc 0 1
⊢ HasFiniteIntegral (fun ω => X i ω ^ 2) μ

Tactic: apply HasFiniteIntegral.of_mem_Icc (a := 0) (b := 1) h_box

State after:

Relevant lemmas:
MeasureTheory.HasFiniteIntegral.of_mem_Icc : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α}
  [inst : IsFiniteMeasure μ] (a b : ℝ) {X : α → ℝ}, (∀ᵐ (ω : α) ∂μ, X ω ∈ Icc a b) → HasFiniteIntegral X μ