Theorems · Theorem · measure theory

Convex.condExp_mem

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {α : Type u_2} {f : α → E}
  {m mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set E} (hm : m ≤ mα)
  [MeasureTheory.SigmaFinite (μ.trim hm)],
  MeasureTheory.Integrable f μ → IsClosed s → Convex ℝ s → (∀ᵐ (a : α) ∂μ, f a ∈ s) → ∀ᵐ (a : α) ∂μ, μ[f | m] a ∈ s

If f lies in a closed convex set s a.e., then μ[f | m] lies in s a.e.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Cited by

0 results in Mathlib

Foundations

Depth 302 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.SigmaFinite