Theorems · Theorem · measure theory

NNRealRMK.integral_rieszMeasure

∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : T2Space X] [inst_2 : LocallyCompactSpace X]
  [inst_3 : MeasurableSpace X] [inst_4 : BorelSpace X] (Λ : CompactlySupportedContinuousMap X NNReal →ₗ[NNReal] NNReal)
  (f : CompactlySupportedContinuousMap X NNReal), ∫ (x : X), ↑(f x) ∂NNRealRMK.rieszMeasure Λ = ↑(Λ f)

The Riesz-Markov-Kakutani representation theorem: given a positive linear functional Λ, the (Bochner) integral of f (as a ℝ-valued function) with respect to the rieszMeasure associated to Λ is equal to Λ f.

Defined in

Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal

Cited by

3 results in Mathlib

Foundations

Depth 262 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceT2SpaceLocallyCompactSpaceMeasurableSpaceBorelSpace