Generator

Definition

For every system of sets $\mathcal{G} \subseteq \mathcal{P}\left. (X) \right.$ there exists a smallest (i.e., minimal) $\sigma$-algebra containing $\mathcal{G}$. This is the $\sigma$-algebra generated by $\mathcal{G}$, denoted by $\sigma\left. \left( \mathcal{G} \right) \right.$, and $\mathcal{G}$ is called its generator (i.e., the smallest $\sigma$-algebra containing a collection of subsets).