An Isomorphism Theorem for Finitely Additive Measures

A problem which is appealing to the intuition in view of the relative frequency interpretation of probability is to define a measure on a countable space which assigns to each point the measure 0. Such a measure of course becomes trivial if it is countably additive. Finitely additive measures of this type have been discussed by R. C. Buck [l] and by E. F. Buck and R. C. Buck [2]. In a discussion of the density of sets of integers, R. C. Buck introduces in [l] a special finitely additive measurable space, D*, containing the arithmetic progressions and assigning to each a measure m* equal to the reciprocal of their period. Special properties of this measure are developed there largely from the number theoretic point of view. In [2] the authors showed that any separable, non atomic, normalized, finitely additive measure was isomorphic to a contraction of [D*, m*]. Necessary and sufficient conditions for a separable, non atomic measure to be point and set isomorphic to the Borel sets have been established by Halmos and von Neumann [3]. Here we consider the analogous problem for a finitely additive measure on a countable space. We show that a wide class of finitely additive measures in a countable space are set isomorphic to Jordan content on [0, 1 ] and point isomorphic to a restriction of Jordan content to a countable dense subset of [0, l]. Let X be a countable space and let up be a real, finitely additive function defined on a countable class P of subsets of X. We assume the following properties of P and pp: (i) Ex, -EjG'P implies EiHEjGP, (ii) Ei, £2GP, £iC£j implies there exists d such that £i = GCC2C • • • CCn=E2 and C—C^GP, (iii) X and 0 are in P and uP(X) = l, pP(0)=0, further EiGP, E^O implies that up(Ei) >0. (iv) For any s£;X there exist EfG<P such that sE£, and