Theorems · Theorem · measure theory

ConcaveOn.le_map_set_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} {t : Set α} {f : α → E} {g : E → ℝ},
  ConcaveOn ℝ s g →
    ContinuousOn g s →
      IsClosed s →
        μ t ≠ 0 →
          μ t ≠ ⊤ →
            (∀ᵐ (x : α) ∂μ.restrict t, f x ∈ s) →
              MeasureTheory.IntegrableOn f t μ →
                MeasureTheory.IntegrableOn (g ∘ f) t μ → ⨍ (x : α) in t, g (f x) ∂μ ≤ g (⨍ (x : α) in t, 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 of a set t to s, then the average value of g ∘ f over t is less than or equal to the value of g at the average value of f over t provided that both f and g ∘ f are integrable.

Defined in

Mathlib.Analysis.Convex.Integral

Cited by

0 results in Mathlib

Foundations

Depth 269 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace