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Nathaniel Virgo (Aug 16 2020 at 00:48):I've been idly wondering about finitely additive probability, which is just probability theory minus the axiom that the probability of a countable union of pairwise disjoint sets has to equal the sum of the probabilities of the individual sets. (Hence, I think, it can be defined in terms of atomic Boolean algebras instead of algebras.) There's a paper about its implications here:
Kadane, Schervish and Seidenfeld (1986) Statistical Implications of Finitely Additive Probability
I'm wondering if it's been studied from the perspective of category-theoretic probability. For example, if one defined a monad of finitely additive probability measures and took its Kleisli category, would that result in a 'nicer' Markov category than the one you get from normal, countably additive probability measures? Given such a category of finitely additive Markov kernels, is there a category-theoretic way to extract the ones that are also countably additive?
It seems like often this kind of generalisation is worthwhile in category theory, so I wondered if this particular one has been studied.