Theorems · Theorem · measure theory

MeasureTheory.setIntegral_mono_on_ae

∀ {X : Type u_1} {E : Type u_3} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : PartialOrder E] [IsOrderedAddMonoid E] [IsOrderedModule ℝ E] {μ : MeasureTheory.Measure X} {f g : X → E}
  {s : Set X} [ClosedIciTopology E],
  MeasureTheory.IntegrableOn f s μ →
    MeasureTheory.IntegrableOn g s μ →
      MeasurableSet s → (∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x) → ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Set

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpacePartialOrderIsOrderedAddMonoidIsOrderedModuleClosedIciTopology