Window count of a point process ($N_W=\Pi (W)$)

For a bounded Borel set $W\subset {\mathbb{R}}^d$, the window count $N_W$ of a locally finite point process is a finite nonnegative integer random variable. It counts multiplicities, and the convention for window boundaries matters. On the time half-line, $N(t)=\Pi ((0,t])$ is a counting process and $N(t)-N(s)=\Pi ((s,t])$. For pairwise disjoint observation windows $W_1,\dots ,W_m$, the measure identityis valid for every realization. Additivity of counts supplies no independence assumption.

Observation windows are measurable sets used to count points of a point process. They are pairwise disjoint when $W_i\cap W_j=\varnothing$ for every $i\ne j$. Half-open intervals such as $[0,1)$ and $[1,2)$ meet at a boundary without sharing any point, whereas the closed intervals $[0,1]$ and $[1,2]$ both contain one. A disjoint union makes window counts additive; it does not make their random variables independent.