Theorems · Theorem · probability

MeasureTheory.Integrable.tendsto_ae_condExp

∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
  [MeasureTheory.IsFiniteMeasure μ] {g : Ω → ℝ},
  MeasureTheory.Integrable g μ →
    MeasureTheory.StronglyMeasurable g →
      ∀ᵐ (x : Ω) ∂μ, Filter.Tendsto (fun n => μ[g | ↑ℱ n] x) Filter.atTop (nhds (g x))

Part c of the L¹ martingale convergence theorem: Given an integrable function g which is measurable with respect to ⨆ n, ℱ n where ℱ is a filtration, the martingale defined by 𝔼[g | ℱ n] converges almost everywhere to g. This martingale also converges to g in L¹ and this result is provided by MeasureTheory.Integrable.tendsto_eLpNorm_condExp

Defined in

Mathlib.Probability.Martingale.Convergence

Cited by

2 results in Mathlib

Foundations

Depth 321 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure