= Solution
The <mapping theorem for Poisson point processes> applies provided $f$ is a <measurable function> and its <pushforward measure>
$$
f_*\Lambda(B)=\Lambda(f^{-1}(B))
$$
is locally finite, meaning
$$
\Lambda(f^{-1}(B))<\infty
$$
for every bounded Borel set $B\subseteq\mathbb R^s$. If “Poisson process” means a simple point process, one also requires $f_*\Lambda$ to be atomless, equivalently
$$
\Lambda(f^{-1}(\{y\}))=0
\qquad(y\in\mathbb R^s).
$$
Under these conditions, $f(\Pi)$ is a <Poisson point process> with mean measure
$$
\boxed{\Lambda_f(B)=f_*\Lambda(B)=\Lambda(f^{-1}(B)).}
$$
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