Baire measure

In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In this context, Baire measures are used because they connect to the properties of continuous functions more directly.

There are several inequivalent definitions of Baire sets, so correspondingly there are several inequivalent concepts of Baire measure on a topological space. These all coincide on spaces that are locally compact σ-compact Hausdorff spaces.

In practice Baire measures can be replaced by regular Borel measures. The relation between Baire measures and regular Borel measures is as follows:

Examples

References

This article is issued from Wikipedia - version of the Monday, August 18, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.