Random measures and locally finite point processes

Fix a locally compact, second-countable Hausdorff topological space $S$ (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Second countability: an at most countable basis for the topology, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). Here locally compact means that every point has a compact neighbourhood; second countable means that the topology has an at most countable basis; and Hausdorff means that distinct points have disjoint open neighbourhoods. Let $\mathcal{B}(S)$ be the Borel sigma-algebra generated by the open sets (The Borel sigma-algebra of a topological space).

A counting measure on $S$ is a Borel measure of the form $\mu ={\sum }_{j\in J}{\delta }_{x_j}$ for a finite or countable index set $J$ and points $x_j\in S$, where repetitions are allowed and the empty sum is zero. Each ${\delta }_x$ is the unit point mass at $x$ (The Dirac set function at a point, A Dirac set function is a probability measure), and the countable sum is a measure by Nonnegative scalar multiples and countable weighted sums of measures are measures. The multiplicity at $x$ is $\mu (\{x\})$. The counting measure is locally finite if $\mu (K)<\mathrm{\infty }$ for every compact subset $K\subseteq S$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Let ${\mathcal{N}}_{\mathrm{l}\mathrm{f}}(S)$ be the set of locally finite counting measures, equipped with the sigma-algebra generated by the evaluation maps $\mu \mapsto \mu (B)$ for $B\in \mathcal{B}(S)$. A point process on $S$ is a measurable map $\eta :(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to ({\mathcal{N}}_{\mathrm{l}\mathrm{f}}(S),\mathcal{E})$ from a probability space (Probability measures and probability spaces), with measurability in the sense of A measurable function between measurable spaces. Equivalently, every count $\eta (B)$ is a measurable random variable, because the evaluation maps generate $\mathcal{E}$. Its law is the probability measure ${\mathbb{P}}_{\eta }(A)=\mathbb{P}({\eta }^{-1}(A))$ on $({\mathcal{N}}_{\mathrm{l}\mathrm{f}}(S),\mathcal{E})$ (Law or distribution of a random element, The law of a random element is a probability measure). The process is classified by its realized configurations: a configuration $\mu$ is simple if $\mu (\{x\})\le 1$ for every $x$, and is multiplicity-bearing if some singleton has mass greater than one. This is a pointwise distinction; no almost-sure simplicity assertion is part of this definition.

Let ${\mathcal{M}}_{\mathrm{l}\mathrm{f}}(S)$ be the set of all locally finite Borel measures on $S$, with the sigma-algebra generated by their evaluation maps. A random locally finite measure $\xi$ is a measurable map into this space.

For a nonnegative Borel function $f$, write $\eta (f):={\int }_{S}fd\eta$. Define the expected measure, or intensity measure, by $\nu (B):=\mathbb{E}[\eta (B)]$. The expected-measure and random-integral measurability assertions are justified by Mean measures and measurable nonnegative integrals of random measures ↗. The measure $\nu$ need not be locally finite; it is locally finite exactly when $\mathbb{E}[\eta (K)]<\mathrm{\infty }$ for each compact $K$.

The mean measure of a random locally finite measure is likewise $\nu (B)=\mathbb{E}[\xi (B)]$, covered by the same expected-measure lemma. Local finiteness of each realization does not by itself assert local finiteness of $\nu$.