Journal of the
Korean Mathematical Society JKMS

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.

Keywords: vector measure, dual Banach space, Riesz characterisation theorem, weakly sequentially complete Banach space, dominated convergence theorem, bounded convergence theorem, function space

MSC numbers: Primary 46G10; Secondary 46E40