\begin{definition}[Definition:Restricted Measure]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$.
Then the '''restricted measure on $\Sigma'$''' or the '''restriction of $\mu$ to $\Sigma'$''' is the mapping $\nu: \Sigma \to \overline \R$ defined by:
:$\forall E' \in \Sigma': \map \nu {E'} = \map \mu {E'}$
That is, $\nu$ is the restriction $\mu \restriction_{\Sigma'}$.
\end{definition}