So in standard modern probability theory we cannot choose uniformly from an infinite set (say a countable set). This is often shown to be the case as we wish to maintain sigma-additivity and that implies the probably of each outcome must be zero, but then that violates the axiom that the entire sample space have probability 1.
That being said, I can "imagine" uniformly choosing, say, a number from $\mathbb N=\{1,2,\ldots\}$. The issue is that no matter what finite set I take, the random number chosen will "almost surely" not be in that set (with "almost surely" taken to intuitively mean here that the probability of any finite set is "VERY small" relative to its complement, since we are already violating the axioms that give meaning to that phrase). In other words, I cannot produce any finite cutoff where I can be comfortable that I have captured the random number. No matter where you look, the random number is "likely" well beyond that bounded region.
So, this idea seems to be at least (potentially) philosophically tractable. Clearly this falls outside the scope of probability theory according to Kolmogorov's axioms. But can we come up with a non-arbitrary list of axioms where choosing uniformly from an infinite set is allowed?
I suppose removing either of the axioms: (1) that the probability of the entire sample space is one, or (2) sigma-additivity for disjoint events, is an obvious option to try. My guess is that removing either of these axioms makes the theory basically useless.
Have any of these situations been discussed anywhere in the literature?
I have seen some discussion of negative probability, maybe in some physics literature, but can't recall exactly where. I don't recall thinking about if that has any impact on uniformly choosing from an infinite set though.
With the standard probability axioms as: For any event $A$ a subset of sample space $\Omega$,
- $P(A)\geq0$
- $P(\Omega)=1$
- $P(\cup A_k)=\sum P(A_k)$ for any at most countable list of pair-wise disjoint events.
If I discard axiom 2, then I lose the relative frequency interpretation of probability maybe. The idea of uniformly choosing from an infinite set seems somewhat tractable, but the sample space just has total probability zero. I can philosophically interpret this as one simply not being able to perform the experiment though, which is obviously true from a physical standpoint. It's not even hypothetically possible at all in some sense. But, mathematically, nevertheless, I can just definite the probability of choosing each natural number as zero, and assume sigma-additivity, and it all seems to work out. Again, presumably not useful for building a bigger applied theory though, but is this at all coherent? Can I assume only axioms 1 and 3, and say "let $x$ be a number chosen uniformly form $\mathbb N$"? I suppose I could just define the probability of each number arbitrarily though. Say, let them each have probability 1. So the probability that $x$ is larger than any arbitrary finite bound is then infinity. That sort of feels better than having them each probability zero, but maybe just as useless.
Any thoughts or references are appreciated.