I note that villegas condition (with monotone continuity) given a total ordering and the other general axioms of qualitative probability are sufficient for a unique strong representation. However villegas original conditional only suggested that one can always find an A subset of B such B>A>empty set; whilst convex spaces require that one can always split the events infinitely many times so that there for probability value smaller than P(A), one can find a B subset of A, such P(B) =epilson for all real numbers A=empty set and likewise for B, and one, where these events are totally ordered; does one need to literally split them up, via the equiprobable event A=B as if its independent product space where for each hypothetical B
althoug in these two papers just atomicity is identified with convexity- although- presumably they mean strongly non-atomic. And that its merely atomless spaces (or which have almost equi-probable partitions) or rather merely strongly continuous spaces which when supplemented with monotone continuity act as if they are convex or strong atomic (unless there is further distinction again between (1) strongly atomless/convex, (2)atomless, and (3) strongly continuous spaces (almost equ-probable partitions, savaves tightness/fine-ness)- where the former requires both monotone continuity in stronger guise as an archimedean constraint (for both countable addivity as a representation, and to neglect infinitesimally probable events for uniqueness, indifference), atomlessness requires it as an archimedean condition for uniqueness, where countable additvity falls (given its finitely representable), and strongly non atomic ones dont require either given they are finitely representable for a countably additive unique representation (presumably countable addivity just falls out, or it can also be applied hypothetically for finite spaces, and there is also no indifference archimedean assumption). I am a little confused. Perhaps absolutely continuous denotes the middle category if there is one at all
Actually the last paper makes it quite clear that non-atomicity and convex ranged are not exactly identical (but only in the presence of countable addivity or some such constraint)- which is presumably why it is sometimes that villegas conditions are little too strong for both uniqueness and representation, given that one can add that the same condition for savages weaker conditions, strongly continuous (tight and fine) I believe, to also yield a measure that acts as if its convex- (I think that this is because savage required uniqueness just to get the representation in infinite spaces; and thus countable addivity or monotone continuity or something was added- that both the representation and the uniqueness stood and fell together)
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