In this paper, X will always denote a completely regular Hausdorff space, K the field of real or complex numbers (we will call them scalars), E a normed space, and Cb(X,E) (resp. Cb(X)) the space of all E-valued (resp. K-valued) bounded continuous functions on X. Also for any topological space Y, C(Y) will denote all K-valued continuous functions on Y. )~ will denote the Stone-Cech compactification. For a continuous: f : X--, Y, Y a topological space, f : 3~ ~ Y will denote its unique continuous extension if possible. For a linear continuous mapping ~:(Cb(X), I1" II)--'K (here I1" It is sup norm), ~ will denote the regular Borel measure on _~ defined by/] : C(X) ~K,/](g) = I~(g/X), for every g ~ cO~). As in [6]/~ can also be considered as a regular finitely additive measure on the algebra generated by zero-sets. Notations of [3] will be used. Ba(X) = Ba, Bo(X) = Bo will respectively denote the a-algebra generated by zero-sets and closed sets in X. For a compact Q c X\X , CQ(X) = { f e Cb(X), ~ = 0 on Q}. The locally convex topology, on Cb(X,E),flQ is generated by semi-norms. II'llh, heCQ(X), Ilfllh --sup IIh(x)f(x)ll. It is the finest locally convex topology agreeing with the x e X topology of uniform convergence on compact subsets of )~\Q, on norm bounded subsets of Cb(X, E) (note for a compact C C X\Q, and f ~ Cb(X, E), s u p f over C is in the sense sup IIf II ~(C)). The topology fl is defined to be