So, I'm trying to get my head around when you can have finitely but not countably additive probabilities.
The standard example of a finitely additive but not countably additive space is the following strange distribution over the natural numbers. All finite sets get measure 0, but the whole space gets measure 1. This is finitely additive but not countably so, since a finite union of finite sets is finite, but a countable union needn't be so.
So this got me thinking that if you had an atomless space, examples of this form would be harder to come by. Does atomlessness plus finite additivity guarantee countable additivity? If not, what is missing?
I know that Villegas (1964) shows that for a comparative probability structure to be countably additive, the important properties of the structure are atomlessness and a certain kind of continuity. But I don't know how relevant that point is to the current question.