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| − | A [[Measure|measure]] defined on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807601.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807602.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807603.png" /> such that for any Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807604.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807605.png" /> there is an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807606.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807608.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r0807609.png" />. An equivalent definition is as follows: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076010.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076011.png" /> there is a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076013.png" />.
| + | #REDIRECT[[Borel measure]] |
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| − | ====Comments====
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| − | See also [[Regular set function|Regular set function]].
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| − | This notion of regular measure should not be confused with that of a regular outer measure, which is an [[Outer measure|outer measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076014.png" /> (cf. also [[Measure|Measure]]) such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076015.png" /> there is a measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080760/r08076017.png" />.
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) pp. 111</TD></TR></table>
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Latest revision as of 09:09, 15 August 2012
How to Cite This Entry:
Regular measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_measure&oldid=13741
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article