= Simple point process
{title2=$\Pi(\{x\})\in\{0,1\}$}
A <point process> is simple when, on a single <almost sure event>, every location has multiplicity at most one. Its realizations are <simple locally finite point measures>. For instance, $\delta_{0.4}+2\delta_{1.1}+\delta_{1.8}$ is locally finite but not simple: it has four points counted with multiplicity at three distinct locations. Simplicity is a statement about each entire realization. Checking $\mathbb P(\Pi(\{x\})>1)=0$ separately for every fixed $x$ is insufficient: $2\delta_U$ with $U$ having the <uniform distribution> on $(0,1)$ passes every such fixed-location check, yet always has a double point.<Lectures on the Poisson Process by Günter Last and others>[Section 6.2, Definitions 6.6 and 6.8, printed p. 49: simple measures and simple point processes]{ref}
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