Theorems · Theorem · measure theory

ConvexOn.map_condExp_le_trim

∀ {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} {mE : MeasurableSpace E}
  [BorelSpace E] (hm : m ≤ mα) [MeasureTheory.SigmaFinite (μ.trim hm)],
  ConvexOn ℝ s φ →
    LowerSemicontinuousOn φ s →
      MeasureTheory.StronglyMeasurable φ →
        (∀ᵐ (a : α) ∂μ, f a ∈ s) →
          IsClosed s →
            MeasureTheory.Integrable f μ → MeasureTheory.Integrable (φ ∘ f) μ → φ ∘ μ[f | m] ≤ᵐ[μ.trim hm] μ[φ ∘ 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

NormedAddCommGroupNormedSpaceCompleteSpaceBorelSpaceMeasureTheory.SigmaFinite