View source for Borel system of sets

''family of Borel sets, or $B$-system, generated by a family of sets $M$''

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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 $M$). 

A standard construction uses transfinite induction, see [[Algebra of sets]].