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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:
 +  {{MSC03E1528A05}} 
 +  [[Category:Descriptive set theory]] 
 +  [[Category:Classical measure theory]] 
 +  {{TEXdone}} 
   
−  i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201403.png" />;
 +  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 setsBorel field of sets]].
 
−   
−  ====References====
 
−  <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.E. Munroe, "Measure and integration" , AddisonWesley (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