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  "body": "\n## Definition\n\nLet $E$ be a complete separable metric space with its Borel $\\sigma$-algebra $\\mathcal{B}(E)$.\nA measure $\\mu$ on $\\mathcal{B}(E)$ is a *locally finite counting measure* if $\\mu(B) \\in \\set{0, 1, 2, \\ldots}$ for every bounded Borel set $B$.\nLet $\\mathcal{N}$ be the set of these measures, with the smallest $\\sigma$-algebra that makes each map $\\mu \\mapsto \\mu(B)$ measurable.\nA *point process* on $E$ is a measurable map $N$ from a [[probability space|0NFPJVJJHX]] $(\\Omega, \\mathcal{F}, P)$ to $\\mathcal{N}$.\n\n## Remarks\n\nSo each $N(B)$, the number of points in $B$, is a [[random variable|0PNRKMRC55]].\nThe point process is *simple* if $N(\\set{x}) \\le 1$ for all $x \\in E$, almost surely.\nIts intensity measure is $\\Lambda(B) = \\mathbb{E}[N(B)]$.\nExample: 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 $\\set{0, \\ldots, n}$.\nNon-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 $[0, 1]$.\nThe [[Poisson point process|0MTNR28K60]] with intensity $\\Lambda$ is the point process with $N(B)$ Poisson of mean $\\Lambda(B)$ and independent counts on disjoint sets.\n",
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