A finite Radon measure satisfies .
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 on the space of continuous functions with compact support (‘Riesz representation definition’). Berg et al. give the following summary , p. 62f.:
is a Radon measure in our sense, while is not always Radon. For locally compact and -compact spaces, however, both coincide (on the Borel algebra) and are equivalent to our Radon measure. For general Hausdorff spaces, Bourbaki introduces , where W, called a Radon premeasure, associates a Radon measure to each compact , with . 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 which are
finite on compact sets, ,
inner regular on the open sets, for open, and compact,
outer regular, for open and Borel .
then the measures correspond bijectively to locally finite Radon measures on .
- 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.
|Date of creation||2013-03-22 15:49:41|
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