The analogy with the generality of vector spaces is actually quite good. It really underscores the fact that, just as in the case of vectors it’s the common axiomatic framework that comes with a variety of concrete models of vector spaces, the relational structure of Kolmogorov’s axioms plays the main role in applications rather than any sort of a unified notion of uncertainty or chance. There are many models of Kolmogorov’s axioms, each comes with its own semantics. More or less what I wrote here: https://realizable.substack.com/p/probabilities-coherence-correspondence.
The last Kolmogorov axiom is usually called countable additivity, not subadditivity. (What's usually called subadditivity would give you an "outer measure," not a measure.) Actually, the restriction to *countable* collections has always struck me as a little bizarre and (a priori) hard to justify. So far as I know, that restriction is only made to get a useful collection of limit theorems for random variables and the expected value: limits of random variables are still random variables, etc. You wouldn't have those if you only had finite additivity.
In the function theory and geometric measure theory that the Kolmogorov axioms are a special case of, the countable additivity is at least legitimized by all the natural examples (Lebesgue measure, Haar measure, Hausdorff measure, etc. or measures with density relative to these) which tend to be the actual objects of study. But I've never seen a similarly convincing case in the context of real-life probability (whatever "real-life probability" might be).
The analogy with the generality of vector spaces is actually quite good. It really underscores the fact that, just as in the case of vectors it’s the common axiomatic framework that comes with a variety of concrete models of vector spaces, the relational structure of Kolmogorov’s axioms plays the main role in applications rather than any sort of a unified notion of uncertainty or chance. There are many models of Kolmogorov’s axioms, each comes with its own semantics. More or less what I wrote here: https://realizable.substack.com/p/probabilities-coherence-correspondence.
The last Kolmogorov axiom is usually called countable additivity, not subadditivity. (What's usually called subadditivity would give you an "outer measure," not a measure.) Actually, the restriction to *countable* collections has always struck me as a little bizarre and (a priori) hard to justify. So far as I know, that restriction is only made to get a useful collection of limit theorems for random variables and the expected value: limits of random variables are still random variables, etc. You wouldn't have those if you only had finite additivity.
In the function theory and geometric measure theory that the Kolmogorov axioms are a special case of, the countable additivity is at least legitimized by all the natural examples (Lebesgue measure, Haar measure, Hausdorff measure, etc. or measures with density relative to these) which tend to be the actual objects of study. But I've never seen a similarly convincing case in the context of real-life probability (whatever "real-life probability" might be).