# Borel system of sets

*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=29890