Theorems · Theorem · measure theory

ConvexOn.map_condExp_le_of_finiteDimensional

∀ {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} [FiniteDimensional ℝ E] (hm : m ≤ mα)
  [MeasureTheory.SigmaFinite (μ.trim hm)],
  ConvexOn ℝ Set.univ φ →
    MeasureTheory.Integrable f μ → MeasureTheory.Integrable (φ ∘ f) μ → φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]

Conditional Jensen's inequality: in a finite dimensional Banach space E with a measure μ that is σ-finite on a sub-σ-algebra m, if φ : E → ℝ is convex, then for any f : α → E such that f and φ ∘ f are integrable, we have φ (𝔼[f | m]) ≤ᵐ[μ] 𝔼[φ ∘ 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

NormedAddCommGroupNormedSpaceCompleteSpaceFiniteDimensionalMeasureTheory.SigmaFinite