Definition:Pushforward Measure

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Definition

Let $\left({X, \Sigma}\right)$ and $\left({X', \Sigma'}\right)$ be measurable spaces.

Let $\mu$ be a measure on $\left({X, \Sigma}\right)$.

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:

$\forall E' \in \Sigma': f_* \mu \left({E'}\right) := \mu \left({f^{-1} \left({E'}\right)}\right)$

where $\overline \R$ denotes the extended real numbers.


$f_* \mu$ is a measure on $\left({X', \Sigma'}\right)$, as shown on Pushforward Measure is Measure.


Also known as

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

Possible other notations for $f_* \mu$ include $f \left({\mu}\right)$ and $\mu \circ f^{-1}$.


Sources