Theorem

restriction of $\sigma$-fields

If $\Omega_0 \subset \Omega$ and $\mathcal{B}$ is a $\sigma$-field of subsets of $\Omega$ then $\{ A \cap \Omega_0 : A \in \mathcal{B} \}$ is a $\sigma$-field of subsets of $\Omega_0$.