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Difference between revisions of "Additive class of sets"

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A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201401.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201402.png" /> satisfying:
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{{MSC|03E15|28A05}}
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[[Category:Descriptive set theory]]
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[[Category:Classical measure theory]]
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i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201403.png" />;
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A terminology used by some authors for [[Ring of sets]]. Correspondingly, a [[Ring of sets|$\sigma$-ring]] is also called completely additive class of sets.
 
 
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201404.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201405.png" />;
 
 
 
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201406.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201407.png" />.
 
 
 
The collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201408.png" /> is a completely additive class of sets if it satisfies:
 
 
 
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201409.png" />;
 
 
 
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014011.png" />;
 
 
 
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014013.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014014.png" />.
 
 
 
A completely additive class is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014016.png" />-field, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a12014018.png" />-algebra or a [[Borel field of sets|Borel field of sets]].
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.E. Munroe,  "Measure and integration" , Addison-Wesley  (1953)  pp. 60</TD></TR></table>
 

Latest revision as of 07:19, 19 September 2012

2010 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A05 [MSN][ZBL]

A terminology used by some authors for Ring of sets. Correspondingly, a $\sigma$-ring is also called completely additive class of sets.

How to Cite This Entry:
Additive class of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_class_of_sets&oldid=28040
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article