Bounded convergence theorems

TL;DR

This paper extends classical bounded convergence theorems to vector-valued function spaces, providing new operator extension results, descriptions of convex set closures, and integral representation theorems for continuous linear operators.

Contribution

It introduces novel extension theorems for linear operators from continuous to measurable vector-valued functions, and offers new insights into convex set closures and vector measure classes.

Findings

Extended bounded convergence theorems for vector-valued functions

New descriptions of convex set closures in function spaces

Strong results on integral representations of operators

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.