\begin{definition}[Definition:Pushforward Measure]
Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $f: X \to X'$ be a $\Sigma \, / \, \Sigma'$-measurable mapping.
Then the '''pushforward of $\mu$ under $f$''' is the mapping $f_* \mu: \Sigma' \to \overline \R$ defined by:
:$\forall E' \in \Sigma': \map {f_* \mu} {E'} := \map \mu {\map {f^{-1} } {E'} }$
where $\overline \R$ denotes the extended real numbers.
\end{definition}