Poisson random measure on a standard Borel space

Let $(E,\mathcal{E})$ be a standard Borel space and let $\nu$ be a sigma-finite measure on it. A counting measure on $(E,\mathcal{E})$ is a measure of the form $\eta ={\sum }_{j\in J}{\delta }_{z_j}$ for a finite or countable index set $J$; repetitions among the points $z_j$ are allowed. Equip the set $\mathcal{N}(E)$ of these measures with the evaluation sigma-algebra generated by the maps $\eta \mapsto \eta (B)$, $B\in \mathcal{E}$. A random counting measure is a measurable map from a probability space into $\mathcal{N}(E)$.

A Poisson random measure with intensity $\nu$ is a random counting measure $N$ such that for every finite pairwise-disjoint family $B_1,\dots ,B_m\in \mathcal{E}$ with $\nu (B_j)<\mathrm{\infty }$, the counts $N(B_1),\dots ,N(B_m)$ are mutually independent and $N(B_j)\sim \mathrm{Pois}(\nu (B_j))$. Here $\mathrm{Pois}(c)$ is the law in Exposure-parameterized Poisson law with exposure $e=c$ and rate $\lambda =1$, so $c=0$ means the point mass at zero. This definition imposes no topology, local compactness, or local-finiteness condition on $E$.