Definition

$\sigma$-field generated by $\mathcal{C}$

Let $\mathcal{C}$ be a collection of subsets of $\Omega$; the $\sigma$-field generated by $\mathcal{C}$ (called $\sigma(\mathcal{C})$) is a $\sigma$-field satisfying

1. $\sigma(\mathcal{C}) \supset \mathcal{C}$
2. if $\mathcal{B}$ is a $\sigma$-field containing $\mathcal{C}$ then $\mathcal{B} \supset \sigma(\mathcal{C})$