BV AS A NON SEPARABLE DUAL SPACE

Let C be a field of subsets of a set I. Also, let ? ?? ? ? ? 1 i i ? be a non-decreasing positive sequence of real numbers such that 1, 1 0 1 ? ? i ? ? and ?? ? ? ? 11 i i ? . In this paper we prove that ?BV of all the games of ?-bounded variation on C is a non-separable and norm dual Banach space of the space of simple games on C . We use this fact to establish the existence of a linear mapping T from ?BV onto F A (finitely additive set functions) which is positive, efficient and satisfies a weak form of symmetry, namely invariance under a semigroup of automorphisms of ?I,C?.