= Point process
A <point process> records random locations by a <random measure> that counts them. Examples include arrival times of a <queue>, positions of <stars>, and the finite <eigenvalue> configuration of a <random matrix>. The count in an observation window is a <random variable>; its <expected value> is supplied by the <intensity measure of a point process>, while its fluctuations and relationships with other windows require more information.
For a precise spatial convention, take $E=\mathbb R^d$ with its <Borel sigma-algebra> and a <probability space> $(\Omega,\mathcal F,\mathbb P)$. A locally finite <point process> $\Pi$ is a <measurable function> from $\Omega$ to the space $\mathcal N_{\mathrm{lf}}(E)$ of locally finite point <measures>, equipped with the <sigma-algebra> generated by evaluations $\nu\mapsto\nu(A)$ for <Borel sets> $A$. Each realization has the form
$$
\Pi_\omega=\sum_{j\in J(\omega)}\delta_{X_j(\omega)},\qquad
\Pi_\omega(A)=\sum_{j\in J(\omega)}\mathbf1_{\{X_j(\omega)\in A\}},
$$
where $J(\omega)$ is finite or <countable> and repeated locations are allowed; $\delta_x$ is a <Dirac measure>. Here a point <measure> is a <counting measure> with possible multiplicities, rather than the <counting measure> of a <set> of distinct locations.<Lectures on the Poisson Process by Günter Last and others>[Section 2.1, Definition 2.1, printed pp. 10-11: point measures and the sigma-algebra generated by evaluations]{ref}
Local finiteness means $\Pi_\omega(B)<\infty$ for every bounded <Borel set> $B$, on a single <almost sure event>. In $\mathbb R^d$ it suffices to check the <countable> collection of balls of integer radius centered at the origin, and this is equivalent to finiteness on every <compact set>. On a general <measurable space> there is no notion of boundedness without additional structure: state a <metric> or an explicit family of observation windows. Another convention on <locally compact spaces> requires finiteness on <compact sets>; this need not coincide with finiteness on all <bounded sets> outside $\mathbb R^d$. Local finiteness does not require finitely many points in all of $E$.<Lectures on the Poisson Process by Günter Last and others>[Section 2.4, Definitions 2.11 and 2.13, printed pp. 16-17: bounded-set local finiteness and the countable ball exhaustion]{ref}
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