Let $X$ be a Hausdorff space. A Borel measure $\mu$ on $X$ is said to be a Radon measure  if it is:

1. 1.

finite on compact sets,

2. 2.

inner regular (tight), $\mu(A)=\sup\{\mu(V)\mid\text{compact}\ V\subset A\}$.

A finite Radon measure satisfies $\mu(A)=\inf\{\mu(G)\mid\text{open}\ G\supset A\}$.

Radon measures are the “most important class of measures  on arbitrary Hausdorff topological spaces” (König , 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 $\mu$ on the space $C^{c}(X)$ of continuous functions with compact support (‘Riesz representation definition’). Berg et al. give the following summary , p. 62f.:

Given the finctional $\mu$ one defines set functions  , in fact Borel measures, the outer measure   $\mu^{*}$ and the essential outer measure $\mu^{\bullet}$ given by

 ${}^{*}(G)$ $\displaystyle=\sup\{\mu(f)|f\in C^{c}(X),0\leq f\leq 1_{G}\}\text{ for open }G% \subset X\text{ and}$ (1) $\displaystyle\mu^{*}(A)$ $\displaystyle=\inf\{\mu^{*}(A\cap K)|K\in\mathfrak{K}(X)\}\text{ for }A\subset X,$ (2) $\displaystyle\mu^{\bullet}(A)$ $\displaystyle=\sup\{\mu^{*}(A\cap K)|K\in\mathfrak{K}(X)\}\text{ for }A\subset X.$ (3)

$\mu^{\bullet}$ is a Radon measure in our sense, while $\mu^{*}$ is not always Radon. For locally compact and $\sigma$-compact spaces, however, both coincide (on the Borel algebra) and are equivalent      to our Radon measure. For general Hausdorff spaces, Bourbaki introduces $W^{*}(A)=\sup\{(W_{K})(A\cap K)|K\in\mathfrak{K}(X)\}$, where W, called a Radon premeasure, associates a Radon measure $W_{K}$ to each compact $K\subset X$, with $W_{K}|L=W_{L},L\in\mathfrak{K}$ . This is a Radon measure (on Borel sets), Bourbaki, however, calls it only so if it is in addition  locally finite.

Consider now Borel measures $\nu:\mathfrak{B}\mapsto[0,\infty]$ which are

• finite on compact sets, $\nu|\mathfrak{K}<\infty$,

• inner regular on the open sets, $\nu(G)=\sup\{\nu(K)|K\subset G\}$ for $G$ open, and $K$ compact,

• outer regular, $\nu(B)=\inf\{\nu(G)|B\subset G$ for open $G$ and Borel $B$.

then the measures $\nu$ correspond bijectively to locally finite Radon measures $\mu$ on $X$.

## References

• 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 RadonMeasure 2013-03-22 15:49:41 2013-03-22 15:49:41 ptr (5636) ptr (5636) 16 ptr (5636) Definition msc 28C05 msc 28C15 BorelMeasure SigmaFiniteBorelMeasureAndRelatedBorelConcepts Radon space