On products of vector measures

Abstract

Products of positive measures play a very important role in analysis. The purpose of this paper is to construct a theory of products of two measures taking values in two (possibly different) Banach spaces. A Fubini theorem is obtained which generalizes the Fubinitheorem for the Bochner integral (Dunford and Schwartz (1958), Theorem 9, page 190), and hence also the classical result.We use the theory of vector integration presented in Dinculeanu (1967). Our arguments rely upon a standard sort of application of the dominated convergence theorem (cf. Dunford and Schwartz (1958), Theorem 9, page 190), and therefore do not appear to generalize to any theory of integration where this theorem is lacking (e.g. Bartle (1956)).