Integral Representation of Continuous Operators with Respect to Strict Topologies

Abstract

Let X be a completely regular Hausdorff space and Bo be the σ-algebra of Borel sets in X. Let C b (X) (resp. B(Bo)) be the space of all bounded continuous (resp. bounded Bo-measurable) scalar functions on X, equipped with the natural strict topology β. We develop a general integral representation theory of (β, ξ)-continuous operators from C b (X) to a lcHs (E, ξ) with respect to the representing Borel measure taking values in the bidual E ξ of (E, ξ). It is shown that every (β, ξ)-continuous operator T : C b (X) → E possesses a (β, ξE)-continuous extensionT : B(Bo) → E ξ , where ξE stands for the natural topology on E ξ. If, in particular, X is a k-space and (E, ξ) is quasicomplete, we present equivalent conditions for a (β, ξ)-continuous operator T : C b (X) → E to be weakly compact. As an application, we have shown that if X is a k-space and a quasicomplete lcHs (E, ξ) contains no isomorphic copy of c0, then every (β, ξ)-continuous operator T : C b (X) → E is weakly compact.