theorem extracted_formal_statement_10.{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) :
  let I := Finset.univ;
  (∀ i ∈ I, Measurable (X i)) → (Measurable fun a => ∑ i ∈ I, X i a) → Measurable fun ω => ∑ i, X i ω := sorry

State before:
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
I : Finset (Fin n) := Finset.univ
hXm' : ∀ i ∈ I, Measurable (X i)
h : Measurable fun a => ∑ i ∈ I, X i a
⊢ Measurable fun ω => ∑ i, X i ω

Tactic: exact h

State after:

Relevant lemmas: