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= Mapping theorem for Poisson point processes
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Let $\Pi$ be a Poisson point process with intensity measure $\mu$ and let $f$ be measurable. If the pushforward
$$
f_*\mu(B)=\mu(f^{-1}(B))
$$
is locally finite, then the image counting measure is Poisson with intensity $f_*\mu$. 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.


 Mapping theorem (point process) (source code)