Vector Measures and Integration Operators

Abstract

It is evident from Chapter 1 that the theory of vector measures and their integration theory play a vital role in this text, both by providing various techniques and as a structural approach in general. Ever since the fundamental monographs [42], [86] appeared some 30 years ago, there has been an ever growing literature on this topic. The first aim of this chapter is to collect together those aspects of the existing theory that are relevant to our needs. Since many of these results are only formulated over real spaces, we will need to extend them (as also done in Chapter 2) to the setting of spaces over ℂ. In some cases this can be achieved by the usual complexification arguments but, for others, entirely new proofs need to be provided. In addition, many new results are also developed and thereby appear for the first time. This is particularly the case in relation to the spaces L^p(v) and L^p/_w(v), for 1 ≤ p ≺ ∞ and v a Banach-space-valued vector measure, and the associated integration operators defined on them. These results are essential for Chapter 5, where we develop the theory of p-th power factorable operators defined on suitable function spaces.