Definition:Pushforward Measure
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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:
- $\forall E' \in \Sigma': \map {f_* \mu} {E'} := \map \mu {f^{-1} \sqbrk {E'} }$
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 $\mu \circ f^{-1}$.
Also see
- Pushforward Measure is Measure, showing that $f_* \mu$ is a measure on $\struct {X', \Sigma'}$.
- Results about pushforward measures can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.6, 7.7$