Abstract Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb (X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · || F ) -continuous operators T : Cb (X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · || F )-continuous operators T : Cb (X, E) → F. As an application, we study (β, || · || F )-continuous weakly compact and unconditionally converging operators T : Cb (X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb (X, E), β) has the V property of Pełczynski.