Lambda BV as a Non Separable Dual Space

Journal article, 2010

Let C be a field of subsets of a set I. Also, let Lambda = {lambda(i)}(i=1)(infinity) be a non-decreasing positive sequence of real numbers such that lambda(1) = 1, 1/lambda(i) -> 0 and Sigma(infinity)(i=1) 1/lambda(i) = infinity. In this paper we prove that Lambda BV of all the games of Lambda-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 Lambda 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).