Poisson and Cox point processes

Let $S$ be as in Random measures and locally finite point processes. A Poisson point process with locally finite, sigma-finite intensity measure $\nu$ is a point process $\eta$ such that for every finite pairwise-disjoint family of Borel sets $B_1,\dots ,B_m$, the counts $\eta (B_i)$ are independent random elements (Independent random elements) and $\eta (B_i)$ has the scalar Poisson law of mean $\nu (B_i)$ (The scalar Poisson count law, Normalization, moments, and generating function of the Poisson count law). If $\nu (B)=\mathrm{\infty }$, the extended count convention is $\eta (B)=+\mathrm{\infty }$ almost surely. Here sigma-finite means that $S$ is a countable union of Borel sets of finite $\nu$-measure (Finite, sigma-finite, and semifinite measures); for the stated $S$, local finiteness also implies sigma-finiteness by the compact exhaustion in Second-countable locally compact Hausdorff spaces admit compact exhaustions.

Let ${\mathcal{M}}_{\mathrm{l}\mathrm{f}}(S)$ be the evaluation-measurable space of locally finite Borel measures. A Cox point process directed by a random measure $\xi$ consists of a point process $\eta$, a random locally finite measure $\xi$, and a supplied probability kernel $K$ from ${\mathcal{M}}_{\mathrm{l}\mathrm{f}}(S)$ to ${\mathcal{N}}_{\mathrm{l}\mathrm{f}}(S)$ such that $K(\xi ,\cdot )$ is a regular conditional law of $\eta$ given $\xi$ (Conditional law given a random element) and, for $P_{\xi }$-almost every $\mu$, $K(\mu ,\cdot )$ is the law of a Poisson point process of intensity $\mu$. The kernel is part of the model data; this definition does not assert that such a kernel exists for every $(\eta ,\xi )$. The mean measure of $\xi$ is $\nu (B)=\mathbb{E}[\xi (B)]$, a measure by Mean measures and measurable nonnegative integrals of random measures.

Both definitions permit multiplicities. They do not assert that diffuse directing measures yield simple processes or that a Cox law determines the law of its directing measure.