What is sigma-algebra?

A $\sigma$-algebra on a set $X$ is a family $\mathcal A$ of subsets containing $\varnothing$, stable under complement and under countable unions (hence countable intersections, set differences, and it contains $X$). The pair $(X, \mathcal A)$ is a measurable space; members of $\mathcal A$ are measurable sets. For any family $\mathcal E$ of subsets, $\sigma(\mathcal E)$ denotes the smallest $\sigma$-algebra containing $\mathcal E$ (the intersection of all of them — an intersection of $\sigma$-algebras is one).