Mapping theorem (point process)

Let ${\displaystyle X,Y}$ be locally compact and polish and let

${\displaystyle f\colon X\to Y}$

be a measurable function. Let ${\displaystyle \mu }$ be a Radon measure on ${\displaystyle X}$ and assume that the pushforward measure

${\displaystyle \nu$ :=\mu \circ f^{-1}}

of ${\displaystyle \mu }$ under the function ${\displaystyle f}$ is a Radon measure on ${\displaystyle Y}$.

Then the following holds: If ${\displaystyle \xi }$ is a Poisson point process on ${\displaystyle X}$ with intensity measure ${\displaystyle \mu }$, then ${\displaystyle \xi \circ f^{-1}}$ is a Poisson point process on ${\displaystyle Y}$ with intensity measure ${\displaystyle \nu$ :=\mu \circ f^{-1}} .[1]