Theorems · Theorem · probability

MeasureTheory.Submartingale.ae_tendsto_limitProcess

∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
  {f : ℕ → Ω → ℝ} {R : NNReal} [MeasureTheory.IsFiniteMeasure μ],
  MeasureTheory.Submartingale f ℱ μ →
    (∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R) →
      ∀ᵐ (ω : Ω) ∂μ, Filter.Tendsto (fun n => f n ω) Filter.atTop (nhds (MeasureTheory.Filtration.limitProcess f ℱ μ ω))

Almost everywhere martingale convergence theorem: An L¹-bounded submartingale converges almost everywhere to a ⨆ n, ℱ n-measurable function.

Defined in

Mathlib.Probability.Martingale.Convergence

Cited by

4 results in Mathlib

Foundations

Depth 318 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure