= Restriction of a measure to a measurable subset
{title2=$\mu|_E(A)=\mu(A),\qquad A\in\mathcal F|_E$}
For a <measure space> $(\Omega,\mathcal F,\mu)$ and $E\in\mathcal F$, the <restriction of a measure to a measurable subset> is the <measure> on $(E,\mathcal F|_E)$ given by the displayed formula. The <trace sigma-algebra> consists of the <measurable sets> contained in $E$. The empty-set axiom and <countable additivity> are inherited directly from $\mu$. An associated <measure> on the original <sigma-algebra> is $\nu(A)=\mu(A\cap E)$ for $A\in\mathcal F$; the two constructions have different underlying <measurable spaces>.
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