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Difference between revisions of "Baire set"

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''in a locally compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150501.png" />''
 
''in a locally compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150501.png" />''
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[[Category:Classical measure theory]]
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{{User:Rehmann/sandbox/MSC|28A05|03E15,54H05}}
  
 
A set belonging to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150502.png" />-ring generated by the class of all compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150503.png" /> that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150504.png" />-sets. A Baire set serves to define the concept of a Baire-measurable function. In all classical particular cases in which measure theory is developed in topological spaces, e.g. in Euclidean spaces, the concept of a Baire set coincides with that of a [[Borel set|Borel set]].
 
A set belonging to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150502.png" />-ring generated by the class of all compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150503.png" /> that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015050/b0150504.png" />-sets. A Baire set serves to define the concept of a Baire-measurable function. In all classical particular cases in which measure theory is developed in topological spaces, e.g. in Euclidean spaces, the concept of a Baire set coincides with that of a [[Borel set|Borel set]].

Revision as of 17:09, 10 January 2012

in a locally compact Hausdorff space

[ 2010 Mathematics Subject Classification MSN: 28A05,(03E15,54H05) | MSCwiki: 28A05   + 03E15,54H05  ]

A set belonging to the -ring generated by the class of all compact sets in that are -sets. A Baire set serves to define the concept of a Baire-measurable function. In all classical particular cases in which measure theory is developed in topological spaces, e.g. in Euclidean spaces, the concept of a Baire set coincides with that of a Borel set.

References

[1] P.R. Halmos, "Measure theory" , v. Nostrand (1950)
How to Cite This Entry:
Baire set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_set&oldid=20208
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article