Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 2

Definition
Let $X$ be a set.

Let $\mathcal G \subseteq \powerset X$ be a collection of subsets of $X$.

The $\sigma$-algebra generated by $\mathcal G$, $\map \sigma {\mathcal G}$, is the intersection of all $\sigma$-algebras on $X$ that contain $\mathcal G$.

Also see

 * Equivalence of Definitions of Sigma-Algebra Generated by Collection of Subsets