Theorems · Theorem · measure theory

ConcaveOn.condExp_map_le_univ

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {α : Type u_2} {f : α → E}
  {φ : E → ℝ} {m mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ mα)
  [MeasureTheory.SigmaFinite (μ.trim hm)],
  ConcaveOn ℝ Set.univ φ →
    UpperSemicontinuous φ →
      MeasureTheory.Integrable f μ → MeasureTheory.Integrable (φ ∘ f) μ → μ[φ ∘ f | m] ≤ᵐ[μ] φ ∘ μ[f | m]

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Cited by

1 results in Mathlib

Foundations

Depth 307 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.SigmaFinite