Conglomerability of probability measures on Boolean algebras

Abstract

For the case where B is a Boolean algebra of events and P is a probability (finitely additive) deFinetti (1972) considered the question of conglomerability of P and found that in many circumstances this natural notion was equivalent to countable additivity of P. Schervish, Seidenfeld, and Kadane (1984) pursued these investigations on the connection between countable additivity and conglomerability in greater detail for the case where B is a σ-algebra. Hill and Lane (1985) and Zame (1988) give alternative proofs. This article is an extension (for the most part) of Schervish, Seidenfeld, and Kadane's work to the case where B is an arbitrary Boolean algebra. The more restrictive notion of positive conglomerability for a class of algebras, including the countable algebras, σ-complete algebras, and inifinite product algebras is treated completely. This class is described by the requirement that a {0, 1}-valued measure be countably additive if every countable family of negligible sets is contained within a negligible set (i.e., corresponds to a P-point of of the Stone space). In general positive conglomerability fails to be equivalent to countable additivity though the degree of failure is minor. Building on techniques of Hill, Lane, and Zame, we obtain partial results on conglomerability for non-σ-complete algebras.