# Definition:Pushforward Measure

## Definition

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.

## Also known as

Some authors call this the image measure of $\mu$ under $f$.

Possible other notations for $f_* \mu$ include $\map f \mu$ and $\mu \circ f^{-1}$.