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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:
| + | {{MSC|03E15|28A05}} |
| | + | [[Category:Descriptive set theory]] |
| | + | [[Category:Classical measure theory]] |
| | + | {{TEX|done}} |
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| − | 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. |
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| − | 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" />;
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| − | 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" />.
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| − | 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:
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| − | a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120140/a1201409.png" />;
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| − | 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" />;
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| − | 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" />.
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| − | 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]].
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.E. Munroe, "Measure and integration" , Addison-Wesley (1953) pp. 60</TD></TR></table>
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Latest revision as of 07:19, 19 September 2012
2020 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=18573
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article