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Difference between revisions of "Field of sets"

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A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200601.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200602.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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{{TEX|done}}
  
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i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200603.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200604.png" />;
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Also  called Boolean algebra or [[algebra of sets|Algebra of sets]]. A collection $\mathcal{A}$ of subsets of some set $X$ which  contains the empty set and is closed under the set-theoretic operations  of finite union, finite intersection and taking complements, i.e. such  that
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* $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
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* $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
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* $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$
  
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ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200605.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200607.png" />.
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If a field of sets is also closed under countable unions then it is called $\sigma$-field, or
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$\sigma$-algebra. We refer the reader to the page [[Algebra of sets]] for more on the topic and for several bibliographic references.
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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200609.png" />-field of sets is a field of sets satisfying in addition
 
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a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006011.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006013.png" />.
 
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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006014.png" />-field is also sometimes called a [[Borel field of sets|Borel field of sets]].
 
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Sometimes, an algebra (respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006015.png" />-algebra) of sets is taken to mean a field (respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006016.png" />-field) of sets.
 
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Loeve,  "Probability theory" , v. Nostrand  (1963)  pp. 59  (Edition: Third)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart&amp;Winston  (1972)  pp. 7</TD></TR></table>
 

Latest revision as of 18:36, 25 November 2012

2020 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A05 [MSN][ZBL]

Also called Boolean algebra or Algebra of sets. A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that

  • $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
  • $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
  • $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$

If a field of sets is also closed under countable unions then it is called $\sigma$-field, or $\sigma$-algebra. We refer the reader to the page Algebra of sets for more on the topic and for several bibliographic references.

How to Cite This Entry:
Field of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Field_of_sets&oldid=28889
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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