Theorems · Theorem · probability

ProbabilityTheory.strong_law_ae_of_measurable

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] [inst_3 : MeasurableSpace E]
  [BorelSpace E] (X : ℕ → Ω → E),
  MeasureTheory.Integrable (X 0) μ →
    MeasureTheory.StronglyMeasurable (X 0) →
      Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) X) →
        (∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) μ μ) →
          ∀ᵐ (ω : Ω) ∂μ,
            Filter.Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) Filter.atTop (nhds (∫ (x : Ω), X 0 x ∂μ))

Preliminary lemma for the strong law of large numbers for vector-valued random variables, assuming measurability in addition to integrability. This is weakened to ae measurability in the full version ProbabilityTheory.strong_law_ae.

Defined in

Mathlib.Probability.StrongLaw

Cited by

1 results in Mathlib

Foundations

Depth 280 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsProbabilityMeasureNormedAddCommGroupNormedSpaceCompleteSpaceMeasurableSpaceBorelSpace