= Solution
For every <Borel set> $A\subseteq\mathbb R^d$, let
$$
N(A)=\#(\Pi\cap A)
$$
be its random point count. A <Non-homogeneous Poisson point process> with <intensity function of a point process> $\lambda$ is characterized by:
* $N(A)$ is finite almost surely for every bounded Borel set $A$;
* for every bounded Borel set $A$,
$$
N(A)\sim\operatorname{Poisson}(\Lambda(A)),
\qquad
\Lambda(A)=\int_A\lambda(x)\,dx;
$$
* the counts $N(A_1),\ldots,N(A_m)$ are <independent random variables> whenever $A_1,\ldots,A_m$ are pairwise disjoint bounded Borel sets.
Because $\Lambda$ has a density with respect to <Lebesgue measure>, it is atomless, so the process is simple: almost surely no location occurs more than once.
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