Theorems · Theorem · measure theory

MeasureTheory.integral_nonneg_of_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] {f : α → E}, 0 ≤ᵐ[μ] f → 0 ≤ ∫ (x : α), f x ∂μ

The integral of a function which is nonnegative almost everywhere is nonnegative.

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Basic

Cited by

17 results in Mathlib

Foundations

Depth 252 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpacePartialOrderIsOrderedAddMonoidIsOrderedModuleClosedIciTopology