I am currently studying the basic theory of Random Finite Sets, a term commonly used in my field to denote simple point processes. These are random variables whose outcomes are finite sets composed of distinguished elements. For example, if the underlying space is $\mathbb{R}$, then $\{\pi, 1, -3/4\}$ is a possible realization of a RFS. On the other hand, $\{\pi,\pi\}$ is not a realization of a RFS.
I must note that my background in measure theory is not particularly strong, which may lead to misconceptions. My question is as follows: Why can't we allow repeated elements in the outcome of an RFS? From a practical point of view, this means that the probability density of an RFS has to be zero over set containing repeated elements.
In other words, what is the benefit of restricting attention to simple point processes?
My Partial Understanding:
I'm uncertain if I'm correct, but it seems to me that we must consider only simple point processes because of the Radon-Nikodym theorem. The issue might be that if we don't assume the point process in question is simple, we cannot characterize it in terms of probability densities. Consequently, we can rely only on probability measures.
Now, if this is true, why does the probability density not exist for a non-simple point process? Is it possible to find a simple counterexample that demonstrates this fact?
I aim to understand this topic as it is connected to a practical problem: I need to probabilistically characterize an RFS that is the union of two independent RFSs. Here, the problem arises because, due to independence, we cannot guarantee distinguished elements in the union. Thus, I want to comprehend why it is so problematic to consider the union of independent RFSs.