Trace sigma-algebra
= Trace sigma-algebra
{title2=$\mathcal F|_E=\{E\cap A:A\in\mathcal F\}$}
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>.