Sigma-algebra generated by a family

For a family $\mathcal C$ of subsets of $X$ , the generated sigma-algebra $\sigma(\mathcal C)$ is the intersection of all sigma-algebras on $X$ containing $\mathcal C$ . This collection is nonempty because it contains the full power set. An intersection of sigma-algebras is a sigma-algebra, so the definition gives the unique smallest one containing $\mathcal C$ .

Generators

To prove that a sigma-algebra contains $\sigma(\mathcal C)$ , it suffices to show it contains every member of $\mathcal C$ . For example, the Borel sigma-algebra is generated by open sets. The generating family need not itself be closed under complements or countable unions.