Let $X$ be a D11: Set such that

(i)	$\mathcal{F} \subseteq \mathcal{P}(X)$ is a D84: Sigma-algebra on $X$
(ii)	$E \subseteq X$ is a D78: Subset of $X$

The sigma-algebra restriction of $\mathcal{F}$ on $X$ with respect to $E$ is the D11: Set \begin{equation} \{ F \cap E : F \in \mathcal{F} \} \end{equation}