Theorems · Theorem · measure theory

ConcaveOn.condExp_map_le

∀ {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} {s : Set E} (hm : m ≤ mα)
  [MeasureTheory.SigmaFinite (μ.trim hm)],
  ConcaveOn ℝ s φ →
    UpperSemicontinuousOn φ s →
      (∀ᵐ (a : α) ∂μ, f a ∈ s) →
        IsClosed s → 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 306 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.SigmaFinite