Theorems · Theorem · measure theory

ConcaveOn.le_map_average

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [CompleteSpace E] {μ : MeasureTheory.Measure α} {s : Set E} {f : α → E} {g : E → ℝ} [MeasureTheory.IsFiniteMeasure μ]
  [NeZero μ],
  ConcaveOn ℝ s g →
    ContinuousOn g s →
      IsClosed s →
        (∀ᵐ (x : α) ∂μ, f x ∈ s) →
          MeasureTheory.Integrable f μ →
            MeasureTheory.Integrable (g ∘ f) μ → ⨍ (x : α), g (f x) ∂μ ≤ g (⨍ (x : α), f x ∂μ)

Jensen's inequality: if a function g : E → ℝ is concave and continuous on a convex closed set s, μ is a finite non-zero measure on α, and f : α → E is a function sending μ-a.e. points to s, then the average value of g ∘ f is less than or equal to the value of g at the average value of f provided that both f and g ∘ f are integrable. See also ConcaveOn.le_map_centerMass for a finite sum version of this lemma.

Defined in

Mathlib.Analysis.Convex.Integral

Cited by

1 results in Mathlib

Foundations

Depth 268 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.IsFiniteMeasureNeZero