Continuous linear functionals on the space of Borel vector measures

Abstract

We study properties of the spaceM of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space M is equipped with the Fortet–Mourier norm ‖ · ‖F and the semivariation norm ‖ · ‖(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space (M, ‖ · ‖F )∗ is contained in but not equal to the space (M, ‖·‖(X))∗. We obtain a representation of the continuous functionals onM in some particular cases. Introduction. This research was inspired by the paper of K. Baron and A. Lasota [1] in which they studied, among other things, properties of continuous (with respect to the Fortet–Mourier norm) linear functionals on the spaceM of vector measures defined on the Borel subsets of a compact metric space X and having values in a real Banach space E. They proved that such a functional φ is represented by an integral, (0.1) φ(μ) = X ψ(x, μ(dx)), for some function ψ : X×E → R satisfying certain conditions (see Section 1 for precise definitions), but they did not know whether the converse was true. We solve this problem under the assumption that E is finite-dimensional. In that case the functional given by (0.1) is continuous with respect to the Fortet–Mourier norm and we obtain another representation of the elements of the space (M, ‖ · ‖F )∗. We also prove that in general every functional given by (0.1) is continuous with respect to the semivariation norm ‖ · ‖(X) and we answer in the negative the natural question whether every functional φ ∈ (M, ‖ · ‖(X))∗ has the form (0.1). The paper is organised as follows. In Section 1 we introduce the terminology and preliminary results used throughout the paper. In Section 2 we 2000 Mathematics Subject Classification: Primary 46E27, 28B05; Secondary 46G10.