What is the definition of locally finite random subset? Reading a reply by Didier, I was wondering 


*

*How is a "random subset" defined? Is
it a measurable mapping, i.e., random
element, from a probability space
to the power set of another set,
with respect to some sigma
algebra(?) defined on the power set?

*How is a random subset being locally
finite defined?


Thanks and regards!
 A: A random subset $\mathcal{N}$ being (almost surely) locally finite means that $\#(\mathcal{N}\cap B)$ is (almost surely) finite for every compact subset $B$ of the target space. In the setting of point processes, one considers only (almost surely) locally finite random subsets $\mathcal{N}$ so, in a way, one avoids at all cost the power set of the target space, which is much too big for measurability purposes. 
The distribution of a  locally finite random subset $\mathcal{N}$ is defined by the distributions of the finite families of integer valued random variables $\#(\mathcal{N}\cap B)$ for compact subsets $B$, just like the distribution of an infinite sequence $(\xi_n)_n$ indexed by the integers is defined by the distributions of the random vectors $(\xi_n)_{n\in I}$ for every finite $I$, aka the marginals of the process. Note in particular that one assumes that $\#(\mathcal{N}\cap B)$ is measurable for every compact $B$. 
When $\#(\mathcal{N}\cap B)$ is almost surely finite for every compact $B$, one can identify the random subset $\mathcal{N}$ with the random measure which puts a unit Dirac mass on every point in $\mathcal{N}$, defined formally bas the unique measure $N$ such that, at least for every measurable bounded function $f$ with bounded support,
$$
\int f(x)\, \mathrm{d}N(x)=\sum_{x\in\mathcal{N}}f(x).
$$
This identification goes through the (trivial) remark that, for every measurable subset $B$, the events $[\mathcal{N}\cap B=\emptyset]$ and $[N(B)=0]$ coincide.
