Theorems · Theorem · measure theory

MeasureTheory.integral_concaveOn_of_integrand_ae

∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {m : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} [inst_2 : PartialOrder E] [IsOrderedAddMonoid E] [IsOrderedModule ℝ E]
  [ClosedIciTopology E] {β : Type u_6} [inst_6 : AddCommMonoid β] [inst_7 : Module ℝ β] {f : α → β → E} {s : Set β},
  Convex ℝ s →
    (∀ᵐ (x : α) ∂μ, ConcaveOn ℝ s (f x)) →
      (∀ a ∈ s, MeasureTheory.Integrable (fun x => f x a) μ) → ConcaveOn ℝ s fun b => ∫ (x : α), f x b ∂μ

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Basic

Cited by

0 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpacePartialOrderIsOrderedAddMonoidIsOrderedModuleClosedIciTopologyAddCommMonoidModule