Sigma-Field Generated by A Families of Sets

Definition

$\forall \mathcal{A}\subset 2^{\Omega },\exists !,\sigma (\mathcal{A})$ Intersections of sigma-fields are a sigma-field, and the Intersection of all Sigma-Field containing $\mathcal{A}$ is the smallest Sigma-Field containing $\mathcal{A}$. Thus, there exist the unique smallest Sigma-Field containing every set in the given Family of Sets $\mathcal{A}$.