I'm trying to better understand the consequences of representing a random set as a
Random element in the space of locally finite closed sets under the Borel sigma algebra generated by the Fell topology
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.