I am trying to understand the definition of point process when reading its Wikipedia article:
Let $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra $B(S)$. Write $\mathfrak{N}$ for the set of locally finite counting measures on $S$ and $\mathcal{N}$ for the smallest σ-algebra on $\mathfrak{N}$ that renders all the point counts
$ \Phi_B : \mathfrak{N} \to \mathbb{Z}_{+}, \varrho \mapsto \varrho(B)$
for relatively compact sets $B$ in $B$-measurable.
A point process on $S$ is a measurable map $ \xi: \Omega \to \mathfrak{N} $ from a probability space $(\Omega, \mathcal F, P)$ to the measurable space $(\mathfrak{N},\mathcal{N})$.
My questions are:
- Is the counting measure the one that gives the cardinality of a measurable subset, as defined in its Wikipedia article? If yes, isn't it that there is only one counting measure on a measurable space, and why in the definition of point process, does "write $\mathfrak{N}$ for the set of locally finite counting measures on $S$" imply that there are more than one counting measures on $S$?
It has been noted[citation needed] that the term point process is not a very good one if S is not a subset of the real line, as it might suggest that ξ is a stochastic process.
Is a point process a stochastic process?
If no, when can it be? How are the two related?
Thanks and regards!