Let $X$ be a set and let $\mathcal{C}$ be a \reftext{def:family-subfamily-subsets-set-2026a}{family} of subsets of $X$.

The intersection of any nonempty collection of \reftext{def:sigma-algebra-measurable-space-2026a}{$\sigma$-algebras} on $X$ is again a $\sigma$-algebra on $X$, since each of the three defining properties is preserved under intersections of families. The family of all subsets of $X$ is a $\sigma$-algebra containing $\mathcal{C}$, so the collection of all $\sigma$-algebras on $X$ containing $\mathcal{C}$ is nonempty.

The \textbf{$\sigma$-algebra generated by $\mathcal{C}$}, denoted $\sigma(\mathcal{C})$, is the intersection of all $\sigma$-algebras on $X$ containing $\mathcal{C}$. It is the smallest $\sigma$-algebra on $X$ containing $\mathcal{C}$: it contains $\mathcal{C}$, and it is contained in every $\sigma$-algebra on $X$ that contains $\mathcal{C}$.