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

1. finite on compact sets,
2. inner regular (tight), $\mu(A) = \sup \{\mu (V) \mid {compact}\ V \subset A\}$ .

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

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

Radon measures are the ``most important class of measures 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 $\mu$ on the space $C^c(X)$ of continuous functions with compact support (`Riesz representation definition'). Berg et al. give the following summary [1], 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

	for open and	(1)
	for	(2)
	for	(3)

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 $B$ .

Bibliography

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.