Finitely additive measures

0. Introduction. The present paper is concerned with real-valued measures which enjoy the property of finite additivity but not necessarily the property of countable additivity. Our interest in such measures arose from two sources. First, the junior author has been concerned with the space of all finitely additive complex measures on a certain family of sets, which under certain conditions can be made into an algebra over the complex numbers. Second, both of us were informed by S. Kakutani of a result similar to our Theorem 1.23, the proof given by Kakutani being different from ours. The problem of characterizing finitely additive measures in some specific way occurred to us as being quite natural, and this problem we have succeeded in solving under fairly general conditions (Theorem 1.22). The body of the paper is divided into four sections. In §1, we consider finitely additive measures in a reasonably general context, obtaining first a characterization of such measures in terms of a countably additive part and a purely finitely additive part. Purely finitely additive measures are then characterized explicitly. In §2, we extend the theorem of Fichtenholz and Kantorovic [i](l) and thereby characterize the general bounded linear functional on the Banach space of bounded measurable functions on a general measurable space. In §3, we consider a number of phenomena which appear in the special case of the real number system. Here we exhibit a number of finitely additive measures which have undeniably curious properties. In §4, we describe connections between our finitely additive measures and certain countably additive Borel measures defined on a special class of compact Hausdorff spaces. We are indebted to Professor S. Kakutani for comments on and improvements in the results obtained. Throughout the present paper, the symbol R designates the real number system, and points are denoted by lower-case Latin letters, sets by capital Latin letters, families of sets by capital script letters. Sets of functions are denoted by capital German letters. For any set X and any A CZX, the characteristic function of A is denoted by \a1. General finitely additive measures. 1.1 Definition. Let X be an abstract set, and let M be a family of subsets of X closed under the formation of finite unions and of complements. Let "M be the smallest family of sets containing Vît and closed under the formation of countable unions and of complements.