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[parent] Radon measure (Definition)

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, $ \mu(A) = \sup \{\mu (V) \mid$   compact$ \ V \subset A\}$.
A finite Radon measure satisfies $ \mu(A) = \inf \{\mu(G) \mid$   open$ \ G \supset A\}$.

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 [1], 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').

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.)

Bibliography

1
Heinz König: Measure and Integration : An Advanced Course in Basic Procedures and Applications.- Berlin, 1997.



"Radon measure" is owned by ptr. [ full author list (2) | owner history (1) ]
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See Also: Borel measure

Also defines:  Radon space
Keywords:  topologiical measure

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Cross-references: subsets, counterexample, metric spaces, locally finite, support, compact, linear functionals, locally compact, Bourbaki, images, continuous, Polish spaces, topological space, inner regular, compact sets, finite, Borel measure, Hausdorff space
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This is version 12 of Radon measure, born on 2006-04-01, modified 2008-06-16.
Object id is 7797, canonical name is RadonMeasure.
Accessed 3085 times total.

Classification:
AMS MSC: 28C05 (Measure and integration :: Set functions and measures on spaces with additional structure :: Integration theory via linear functionals , representing set functions and measures)
 28C15 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological spaces )
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