Poisson mapping theorem ($f_{\ast }\Pi \sim \mathrm{PRM}(f_{\ast }\nu )$)

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.

For unit volume intensity in the unit ball of ${\mathbb{R}}^d$, polar coordinates give intensity $r^{d-1}dr\sigma (d\omega )$. The separate radius and direction images are Poisson processes, but each counts the same random total number of original points. Their count covariance is the displayed intensity of the intersection of their two preimages. Conditional on the total count, radii and directions are independent samples, which does not imply unconditional independence of the two point processes.