Lebesgue Spaces for Bilinear Vector Integration Theory

In this note we shall announce results concerning the structure of £^0^)> the space of 2s-valued functions integrable with respect to a measure m:H->L(E, F), where L(E,F) is the class of bounded operators from the Banach space E into the Banach space F. The bilinear integration theory introduced here is more restrictive than the one developed by Bartle [1], but it is general enough to allow a norm to be denned on the integrable functions and to permit the study of weak compactness and convergence theorems; moreover, LE(m) lends itself in a natural way to the study of continuous operators T: CE(S)->F, where the domain is the space of continuous is-valued functions defined on the compact Hausdorff space S as follows : By Dinculeanu's representation theorem [6], there exists a unique regular finitely-additive measure m : 2—•£(£", F* *), where 2 is the family of Borel subsets of S, such that T(f)=$fdm. If Tis a weakly compact operator, Brooks and Lewis [2] have shown that m is countably additive, with range in L(E, F). In addition, the set N={\mz\:z e F*} is relatively weakly compact in ca(2)—here mz is the 2s*-valued measure defined by mz(A)e=(m(A)e, z), and \mz\ is the total variation function of mz. Conversely, if N has the above property and E is reflexive, then T is weakly compact. A natural question is whether a Lebesgue space LE(m)=> CE(S) of m-integrable functions can be defined. If so, what convergence theorems can be proved, and how are the weakly compact sets characterized ? The setting is as follows. Let S be a cr-algebra of subsets of a set T, and ra:2-*L(is, F), a countably additive measure be given such that m is strongly bounded, that is, mE ^ ( /Q-^0 , whenever (A{) is a disjoint sequence of sets (mE F is the semivariation of m with respect to E and F [6]). It follows that N={\mz\:zeF*} is relatively weakly compact in ca(S). Let X be a positive control measure for m such that X^.mE E and