Let ,X, be a completely regular Hausdorff space and ,mathcal {B}o, be the sigma -algebra of Borel sets in X. Let C_b(X) (resp. B(mathcal {B}o)) be the space of all bounded continuous (resp. bounded mathcal {B}o-measurable) scalar functions on X, equipped with the natural strict topology beta . We develop a general integral representation theory of (beta ,xi )-continuous operators from C_b(X) to a lcHs (E,xi ) with respect to the representing Borel measure taking values in the bidual E''_xi of (E,xi ). It is shown that every (beta ,xi )-continuous operator T:C_b(X)rightarrow E possesses a (beta ,xi _mathcal {E})-continuous extension {hat{T}}:B(mathcal {B}o)rightarrow E''_xi , where xi _mathcal {E} stands for the natural topology on E''_xi . If, in particular, X is a k-space and (E,xi ) is quasicomplete, we present equivalent conditions for a (beta ,xi )-continuous operator T:C_b(X)rightarrow E to be weakly compact. As an application, we have shown that if X is a k-space and a quasicomplete lcHs (E,xi ) contains no isomorphic copy of c_0, then every (beta ,xi )-continuous operator T:C_b(X)rightarrow E is weakly compact.