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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171802.png" />-system, generated by a system of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171803.png" />'' | + | ''family of Borel sets, or $B$-system, generated by a family of sets $M$'' |
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| − | The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171804.png" />-system of sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171805.png" />, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171806.png" />. The sets belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171807.png" /> are called the Borel sets (cf. [[Borel set|Borel set]]), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b0171809.png" />-sets, generated by the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718010.png" />. For each ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718012.png" /> is the initial ordinal number of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718013.png" />, the Borel classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718014.png" /> are defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718015.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718016.png" /> consists of the unions if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718017.png" /> is odd, and it consists of the intersections if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718018.png" /> is even, of sequences of sets belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718020.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718021.png" />. The same Borel system of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718022.png" /> is obtained if the union and intersection above are interchanged. A Borel set belongs properly to a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718023.png" /> if it belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718024.png" /> but does not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718026.png" />. (Sometimes one takes non-intersecting classes, i.e. then the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017180/b01718027.png" /> is called a class.)
| + | {{MSC|28A05}} |
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| | + | It is seldom used for [[Borel field of sets]] or [[Algebra of sets#sigma-algebra|$\sigma$-algebra]] generated by the set $M$, i.e. the smallest $\sigma$-algebra of subsets of a given set $X$ containing a given family $M$. |
| | + | It is also sometimes used by other authors to denote the smallest family $\mathcal{A}$ containing $M$ which is closed under countable unions and countable intersections (since it is not required that $\mathcal{A}$ be closed under taking complements, $\mathcal{A}$ might be strict subfamily of the $\sigma$-algebra generated by $M$). |
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| | + | A standard construction uses transfinite induction, see [[Algebra of sets]]. |
Latest revision as of 14:02, 5 July 2013
family of Borel sets, or $B$-system, generated by a family of sets $M$
2020 Mathematics Subject Classification: Primary: 28A05 [MSN][ZBL]
It is seldom used for Borel field of sets or $\sigma$-algebra generated by the set $M$, i.e. the smallest $\sigma$-algebra of subsets of a given set $X$ containing a given family $M$.
It is also sometimes used by other authors to denote the smallest family $\mathcal{A}$ containing $M$ which is closed under countable unions and countable intersections (since it is not required that $\mathcal{A}$ be closed under taking complements, $\mathcal{A}$ might be strict subfamily of the $\sigma$-algebra generated by $M$).
A standard construction uses transfinite induction, see Algebra of sets.
How to Cite This Entry:
Borel system of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_system_of_sets&oldid=16701
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article