 Solution

Let $\Pi$ be a Poisson point process on ${\mathbb{R}}^d$ with intensity measurewhere $\lambda$ is nonnegative and locally integrable. Let $f:{\mathbb{R}}^d\to {\mathbb{R}}^s$ be measurable and suppose its pushforward measureis locally finite. The mapping theorem for Poisson point processes says that the image counting measureis a Poisson random measure with intensity $\nu$.

For $f(\Pi )$ to be a spatial Poisson process in the usual simple sense, one also assumes that $\nu$ is diffuse:This prevents collisions with positive probability. If $\nu$ is absolutely continuous, its Radon--Nikodym derivative $\tilde{\lambda }$ is the image intensity function, characterized by