Poisson mapping theorem
= Poisson mapping theorem
{c}
{title2=$f_*\Pi\sim\operatorname{PRM}(f_*\nu)$}
For a <measurable> map $f$ and a <Poisson random measure> of intensity $\nu$, image counts in disjoint sets equal original counts in their disjoint preimages. They are therefore <independent> Poisson variables with the pushforward intensity. The result is a <Poisson random measure> even if images collide; diffuse pushforward intensity gives a simple point process.