Theorems · Theorem · measure theory

ConvexOn.average_mem_epigraph

∀ {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],
  ConvexOn 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), f x ∂μ, ⨍ (x : α), g (f x) ∂μ) ∈ {p | p.1 ∈ s ∧ g p.1 ≤ p.2}

Defined in

Mathlib.Analysis.Convex.Integral

Cited by

3 results in Mathlib

Foundations

Depth 266 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.IsFiniteMeasureNeZero