Trace sigma-algebra ($\mathcal{F}{|}_E=\{E\cap A:A\in \mathcal{F}\}$)

For a measurable space $(\Omega ,\mathcal{F})$ and any subset $E\subseteq \Omega$, the trace sigma-algebra is the displayed family of subsets of $E$. It contains $E=E\cap \Omega$, is closed under relative complements since $E\setminus (E\cap A)=E\cap (\Omega \setminus A)$, and is closed under countable unions. If $E\in \mathcal{F}$, it equals $\{B\in \mathcal{F}:B\subseteq E\}$. Without that measurability assumption, the traces need not be measurable sets in the original sigma-algebra.