Theorems · Theorem · measure theory

MeasureTheory.integral_monotoneOn_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 : Preorder β] {f : α → β → E} {s : Set β},
  (∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s) →
    (∀ a ∈ s, MeasureTheory.Integrable (fun x => f x a) μ) → MonotoneOn (fun b => ∫ (x : α), f x b ∂μ) s

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Basic

Cited by

0 results in Mathlib

Foundations

Depth 254 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpacePartialOrderIsOrderedAddMonoidIsOrderedModuleClosedIciTopologyPreorder