Theorems · Theorem · probability

MeasureTheory.tendsto_eLpNorm_condExp

∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
  [MeasureTheory.IsFiniteMeasure μ] (g : Ω → ℝ),
  Filter.Tendsto (fun n => MeasureTheory.eLpNorm (μ[g | ↑ℱ n] - μ[g | ⨆ n, ↑ℱ n]) 1 μ) Filter.atTop (nhds 0)

Lévy's upward theorem, L¹ version: given a function g and a filtration ℱ, the sequence defined by 𝔼[g | ℱ n] converges in L¹ to 𝔼[g | ⨆ n, ℱ n].

Defined in

Mathlib.Probability.Martingale.Convergence

Cited by

0 results in Mathlib

Foundations

Depth 323 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure