# Measurability of a point process or random measure at a measurable subset

1. 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?

2. More generally, same questions for a random measure instead of a point process?
3. 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!

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I was wondering, did you try to read a classic on Poisson point processes, such as Poisson Processes by J. F. C. Kingman amazon.com/Poisson-Processes-Oxford-Studies-Probability/dp/…;, which is very welcoming and not so long, or another reference given on the linked page? –  Did Apr 22 '11 at 0:31