I am currently reading Point Processes and Their Statistical Inference (2nd Edition) and had a question about how point processes are defined on page 5.
So a point process on $E$ is defined as a measurable mapping $N$ of $(\Omega,\mathcal{F})$ into $(\mathbf{M}_p,\mathcal{M}_p)$ where
- $\mathbf{M}=$ set of all Radon measures
- $\mathbf{M}_p=\{\mu\in \mathbf{M}\;| \; \mu(A) \in \mathbb{N} \; \forall \; A \in \mathcal{B}\}$
- $\mathcal{B}$ the ring of bounded Borel sets
- $\mathcal{M}_p$ the trace sigma algebra on $\mathbf{M}_p$
Point processes are then described as follows. "A point process $N$ is a random distribution of indistinguishable points in $E$: $N(A)$ is the number of points in the set $A$". Earlier in the chapter it says that "a point process is a model of points randomly distributed in some space; they are indistinguishable except for their locations".
I am trying to understand what is meant here by "indistinguishable" since generally when you see a realization of a point process it gives you a set of points that are distinct. Wikipedia says two points are indistinguishable is they have the same neighbourhood, so to me it seems it should be a distribution of distinguishable points in $E$.
For clarification: I am using the term element when talking about an element of a set, and the term point, when talking about a unit of measure assigned to a set by a point process.
When $E$ is an abstract topological space, it could be that two elements — as in „two distinct elements of the topological space“ — are not distinguishable with respect to the topology. This is the case, if any neighbourhood of one point is also a neighbourhood of the other.
To construct an example, let $E= \{x,y\}$, and let the topology be $O= \{\emptyset, E\}$. Then $E$ is the only neighbourhood of each element, and both elements have the same neighbourhoods.
A point process now randomly assigns mass to the open sets (only E) of the topology. Let‘s say for example, that in one realization, it assigns the value 2 to $E$. Then we know, that there are two points in the set $\{x,y\}$. But we don‘t know, if the two points are $x$ and $y$, or maybe $x$ twice, or $y$ twice. We only know that two points in $\{x,y\}$ were randomly chosen.
Note: One often assumes the topology on $E$ to be rich enough so that one can distinguish all the elements. Any Hausdorff space would do just fine.