Point process

Remarks

So each $N(B)$, the number of points in $B$, is a random variable. The point process is simple if $N(\set{x}) \le 1$ for all $x \in E$, almost surely. Its intensity measure is $\Lambda(B) = \mathbb{E}[N(B)]$. Example: if $U_1, \ldots, U_n$ are random points of $[0, 1]$, then $N = \sum_{i=1}^n \delta_{U_i}$ is a point process; $N(B) = \#\set{i : U_i \in B}$ is a random variable with values in $\set{0, \ldots, n}$. Non-example: $\sum_{k \ge 1} \delta_{1/k}$ is not a locally finite counting measure on $\R$, since it gives infinite mass to the bounded set $[0, 1]$. The Poisson point process with intensity $\Lambda$ is the point process with $N(B)$ Poisson of mean $\Lambda(B)$ and independent counts on disjoint sets.