Bounded convergence theorems

Abstract: There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions in a way such that uniformly bounded sequences of functions that converge pointwise in the weak (or norm) topology of E are sent to sequences that converge in the weak, norm or weak* topology of the target space. As an application, a new description of uniform closures of convex subsets of C(X,E) is given. Also new and strong results on integral representations of continuous linear operators defined on C(X,E) are presented. A new classes of vector measures are introduced and various bounded convergence theorems for them are proved.