Projection in Certain Spaces of Set Functions

Let (I, 𝒞) denote a measurable space which is isomorphic to the space ([0, 1], ℬ) where ℬ denotes the collection of Borel subsets of [0, 1]. The space BV consists of all real valued functions u on 𝒞 of the form u = u⁺ − u⁻ where u⁺ and u⁻ are increasing. It is first shown that BV is a dual space. We use this fact to establish the existence of a linear mapping T from BV onto FA (finitely additive set functions) which is positive, efficient and satisfies a weak form of symmetry, namely in variance under a semigroup of automorphisms of (I, 𝒞).