= Window count of a point process
{title2=$N_W=\Pi(W)$}
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> $W_1,\ldots,W_m$, the <measure> identity
$$
\Pi\!\left(\bigcup_{i=1}^m W_i\right)=\sum_{i=1}^m\Pi(W_i)
$$
is valid for every realization. Additivity of counts supplies no <independence> assumption.
\Image[/point-process-window-counts-and-multiplicity.png]
{title=Window counts include multiplicity and depend on boundary conventions}
{description=One locally finite point <measure> is $\delta_{0.4}+2\delta_{1.1}+\delta_{1.8}+\delta_{2.4}+\delta_{3.0}$. The half-open windows $A=[0,1.5)$ and $B=[1.5,3)$ have counts three and two; $\Pi(A\cup B)=5$. The atom at three lies outside both windows. Two stacked dots denote the double atom at $1.1$, and the step plot records $N(t)=\Pi((0,t])$, which includes the atom at three.}
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