 Mapping theorem for Poisson point processes

Let $\Pi$ be a Poisson point process with intensity measure $\mu$ and let $f$ be measurable. If the pushforwardis locally finite, then the image counting measure is Poisson with intensity $f_{\ast }\mu$. If one requires a simple point process, the pushforward must also be diffuse, so distinct source points do not produce coincident image points with positive probability.