Especifying a measurable space for a homogeneos Poisson process.

Question

I'm studying about the Poisson process (PP), and so far I can not find anything about the measurable space that the PP is defined.

Then, I would like to know what is the measurable space for a homogeneous PP, can anyone give me some reference or show what it could be?

Answer 1

Short answer is the following: A point process is a random variable on the set of locally finite counting measures. The sigma algebra defined on this set of locally finite counting measures is the smallest simga algebra, such that the random variables $N_B$ are measurable. Let $\eta$ a locally finite counting measure, then $N_B$ is defined as $$ N_B(\eta) := \eta(B)$$ for all Borel sets $B$.

For a longer answer, have a look in the reference: http://www.math.kit.edu/stoch/~last/seite/lectures_on_the_poisson_process/media/lastpenrose2017.pdf