Suppose $\xi$ is a point process on $(S, B(S))$, where $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra $B(S)$.
I was wondering if $\xi(A), \forall A \in B(S)$ is measurable, i.e. an integer-valued random variable? So that, for example, it makes sense to talk about $ E ( \xi(A) )$? If not, are there some cases when it is true?
- More generally, same questions for a random measure instead of a point process?
- If the answers to the above questions are yes, can we say a point process or random measure is a stochastic process with index set being the $\sigma$-algebra $B(S)$?
Thanks and regards!