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.