= Solution
A <Non-homogeneous Poisson point process> with <intensity function of a point process>[intensity function] $\lambda$ is a random counting measure $N$ such that, for every finite family of pairwise disjoint bounded Borel sets $A_1,\ldots,A_k$, the counts $N(A_1),\ldots,N(A_k)$ are <independent random variables> and
$$
N(A_i)\sim\operatorname{Poisson}\left(\int_{A_i}\lambda(x)\,dx\right).
$$
The assumed local integrability makes each <Poisson distribution>[Poisson] parameter finite. This count law and independence characterize the process.
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