Fell topology versus vague topology for representing random sets

Question

I'm trying to better understand the consequences of representing a random set as a

1. Random element in the space of locally finite closed sets under the Borel sigma algebra generated by the Fell topology

2. Random element in the closed subset of locally finite integer-valued measures under the Borel sigma algebra generated by the (trace of the) vague topology.

My intuition says that the sigma algebra in #2 contains that of #1, but perhaps they are even equivalent?

In terms of continuous functions, if I understand correctly, it seems an obvious gap is that in #2, the count function is continuous taking a (bounded continuity) set to its cardinality is continuous, whereas I believe that the same function is merely lower semicontinuous in the Fell topology... or perhaps I've gotten confused.

Answer 1

Your intuition is correct at least in the case of locally compact second countable $S$. The map $\mu \to supp \, \mu$ is continuous from locally finite integer valued Borel measures on $S$ in the vague topology to closed subsets of $S$ in the Fell topology. If $\mu_n \to \mu$ vaguely and $U$ is open in $S$ with $supp \, \mu \cap U \neq \emptyset$ then $\mu (supp \, \mu \cap U) > 0$ and since $\mu$ is integer valued there exists $x \in U$ such that $\mu \{x \} > 0$ hence $\mu_n \{ x \} > 0$ eventually yielding $supp \, \mu_n \cap U \neq \emptyset$ eventually. A similar argument with compact $K$ satisfying $supp \, \mu \cap K = \emptyset$ finishes the proof. To see that vague topology is finer simply take a point $x \in S$ and consider $n \delta_x$ which doesn't converge vaguely but the support is constant. I'm not sure what the story is if we restrict ourselves to integer valued measures without multiplicity.

Of course none of the above is probabilistic but continuous mapping shows that for random integer valued measures (point processes) with $\mu_n$ converging to $\mu$ in distribution, the random sets $supp \, \mu_n$ converge to $supp \, \mu$ in distribution. Chapter 16 of Kallenberg shows that $supp \, \mu$ in distribution and $E \mu_n \to E \mu$ vaguely implies $\mu_n \to \mu$ in distribution.