Restriction of a measure to a measurable subset ($\mu {|}_E(A)=\mu (A),A\in \mathcal{F}{|}_E$)

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.