Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series A, Pt. 3, pp. 476-491

NON-CONGLOMERABILITY FOR FINITE-VALUED, FINITELY ADDITIVE PROBABILITY

By

T. SEIDENFELD, M.J. SCHERVISH and J.B. KADANE, *Carnegie Mellon University, Pittsburgh*

*SUMMARY. *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, [inf_{he pi}
*P*(*E*|*h*), sup_{he p}*p*(*E*|*h*)], taken from a countable
partition p = h_{j}:*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 *U*_{P}. We show that when ultrafilter *U*_{P} 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 *U*_{P} is
Rudin-Keisler minimal, then *P* is simultaneously conglomerable in
each finite collection of partitions, though not simultaneously
conglomerable in all partitions.

*AMS (1991) subject classification. *60A99, 04A20.

*Key words and phrases. *finitely-additive probability,
non-conglomerability, non-principal ultrafilter, Ramsey ultrafilter,
Rudin-Keisler partial order, ultrafilter