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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" />''
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''in a locally compact Hausdorff space $X$''
  
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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]].
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{{MSC|28A05|03E15,54H05}}
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[[Category:Classical measure theory]]
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A set belonging to the $\sigma$-ring generated by the class of all compact sets in $X$ that are $G_\delta$-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]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , v. Nostrand (1950)</TD></TR></table>
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|valign="top"|{{Ref|H}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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Latest revision as of 15:04, 1 May 2014

in a locally compact Hausdorff space $X$

2020 Mathematics Subject Classification: Primary: 28A05 Secondary: 03E1554H05 [MSN][ZBL]

A set belonging to the $\sigma$-ring generated by the class of all compact sets in $X$ that are $G_\delta$-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

[H] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
How to Cite This Entry:
Baire set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_set&oldid=15423
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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