Suppose I have multiple distinct and disjoint and mutually exclusive sample space, either actual or hypothetical. Where each space is individually finitely representable for instance, but where one can make probability comparisons between the said spaces. Are there uniqueness theorems in existence that allows one to use these different space to fine grain them so as to see if one can derive a unique representation on any or rather for all individual spaces collectively. It would be as if, its a product spaces, although I am considering multiple hypothetical and actual distinct spaces (of which there uncountably many, one for each possible probability value on any given space, as if the more probable events on the other space are supsets of its actual self when less probable), where there are logical relations between these spaces which induce comparison relations.
I believe that sometimes this is done. so that a uniqueness result can be arrived at for a finite space, as if it were atomless.
Perhaps, this was how gleason derived his uniqueness theorem, although I do not know. Does one literally have to use the different space to split the events on any given space, in twine, so as to find a middle point; or to derive that convexity; and if so what other uniqueness theorems are out there other than those which use splitting, convexity, atom-less, when there infinitely many relations. Perhaps some kind of strong continuity or archimedean condition would do it (over and above monotone continuity; as I am not sure how one can really use addivity between spaces)?, but I may be able to without it; although I am not sure. I can post up a diagram of what I mean, or rather multiple of them. Likewise, would scotts 'infinite axiom' need to be explicitly deduced in this case (if at all possible). Or would something Like Luce's condition L work; (there appears to be some kind of symmetric difference that I have noticed between the relations induced by my space; and I believe there may be a uniqueness theorem that covers these kind of cases).
Likewise my 'disjoint' equiprobable are modelled as if they are logical equivalents; prA=~peA iff a full belief that A is true iff a full belief that ~A is true. Would a standard representation be relevant, or could one just use truth-tables to model the possible worlds, despite the fact that 'in some sense', equi-probable events occur in the same possible world in my modal logic. It might seem a bit contradictory otherwise if I do,s
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