Theorems · Theorem · measure theory

MeasureTheory.Integrable.uniformIntegrable_condExp

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_2} [MeasureTheory.IsFiniteMeasure μ]
  {g : α → ℝ},
  MeasureTheory.Integrable g μ →
    ∀ {ℱ : ι → MeasurableSpace α}, (∀ (i : ι), ℱ i ≤ m0) → MeasureTheory.UniformIntegrable (fun i => μ[g | ℱ i]) 1 μ

Given an integrable function g, the conditional expectations of g with respect to a sequence of sub-σ-algebras is uniformly integrable.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Cited by

1 results in Mathlib

Foundations

Depth 313 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure