Radon measure

Let X be a Hausdorff space. A Borel measure μ on X is said to be a Radon measureMathworldPlanetmath if it is:

  1. 1.

    finite on compact sets,

  2. 2.

    inner regular (tight), μ(A)=sup{μ(V)compactVA}.

A finite Radon measure satisfies μ(A)=inf{μ(G)openGA}.

Radon measures are not necessarily locally finitePlanetmathPlanetmath, although this is the case for locally compact and metric spaces. (CounterexampleMathworldPlanetmath: spaces where only finite subsets are compact.)

A Radon space is a topological spaceMathworldPlanetmath on which every finite Borel measure is a Radon measure, this is the case, e.g. for Polish spacesMathworldPlanetmath or Hausdorff spaces that are continuousMathworldPlanetmathPlanetmath images of Polish spaces.

Radon measures are the “most important class of measuresMathworldPlanetmath on arbitrary Hausdorff topological spaces” (König [2], p.xiv) and formed the base of the development of integration theory by Bourbaki and Schwartz. In particular for locally compact spaces one often defines Radon measures as linear functionals μ on the space Cc(X) of continuous functions with compact support (‘Riesz representation definition’). Berg et al. give the following summary [1], p. 62f.:

Given the finctional μ one defines set functionsMathworldPlanetmath, in fact Borel measures, the outer measureMathworldPlanetmathPlanetmath μ* and the essential outer measure μ given by

(G)* =sup{μ(f)|fCc(X),0f1G} for open GX and (1)
μ*(A) =inf{μ*(AK)|K𝔎(X)} for AX, (2)
μ(A) =sup{μ*(AK)|K𝔎(X)} for AX. (3)

μ is a Radon measure in our sense, while μ* is not always Radon. For locally compact and σ-compact spaces, however, both coincide (on the Borel algebra) and are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to our Radon measure. For general Hausdorff spaces, Bourbaki introduces W*(A)=sup{(WK)(AK)|K𝔎(X)}, where W, called a Radon premeasure, associates a Radon measure WK to each compact KX, with WK|L=WL,L𝔎 . This is a Radon measure (on Borel sets), Bourbaki, however, calls it only so if it is in additionPlanetmathPlanetmath locally finite.

Consider now Borel measures ν:𝔅[0,] which are

  • finite on compact sets, ν|𝔎<,

  • inner regular on the open sets, ν(G)=sup{ν(K)|KG} for G open, and K compact,

  • outer regular, ν(B)=inf{ν(G)|BG for open G and Borel B.

then the measures ν correspond bijectively to locally finite Radon measures μ on X.


  • 1 Christian Berg, Jens Peter Reus, Paul Ressel: Harmonic analysis on semigroups. – Berlin, 1984 (Graduate Texts in Mathematics; 100)
  • 2 Heinz König: Measure and integration : an advanced course in basic procedures and applications.– Berlin, 1997.
Title Radon measure
Canonical name RadonMeasure
Date of creation 2013-03-22 15:49:41
Last modified on 2013-03-22 15:49:41
Owner ptr (5636)
Last modified by ptr (5636)
Numerical id 16
Author ptr (5636)
Entry type Definition
Classification msc 28C05
Classification msc 28C15
Related topic BorelMeasure
Related topic SigmaFiniteBorelMeasureAndRelatedBorelConcepts
Defines Radon space