Non-Conglomerability for Finite-Valued, Finitely Additive Probability

We consider how an unconditional, finite-valued, finitely additive probability P on a countable set may localize its non-conglomerability (non-disintegrability). Non-conglomerability, a characteristic of merely finitely additive probability, occurs when the unconditional probability of an event P(E) lies outside the closed interval of conditional probability values, [infhe pi P(E|h), suphe pp(E|h)], taken from a countable partition p = hj:j=1,...}. The problem we address is how to identify events and partitions where a finite-valued, finitely additive probability fails to satisfy conglomerability. We focus on the extreme case of 2-valued finitely additive probabilities that are not countably additive. These are, equivalently, non-principal ultrafilters. Evidently, the challenge we face is that given a countable partition, at most one of its elements has positive probability under P. Thus, we must find ways of regulating the coherent conditional probabilities, given null events, that cohere with the unconditional probability P. Our analysis of P proceeds by the use of combinatorial properties of the associated non-principal ultrafilter UP. We show that when ultrafilter UP is not minimal in the Rudin-Keisler partial order of b(w)\w, we may locate a partition in which P fails to satisfy the conglomerability principle by examining (at most) countably many partitions. This result is then applied to finitely additive probabilities that assume only finitely many values. By contrast, if ultrafilter UP is Rudin-Keisler minimal, then P is simultaneously conglomerable in each finite collection of partitions, though not simultaneously conglomerable in all partitions.