Let Crc = CrC(X, E) denote the space of all continuous functions f, from a completely regular Hausdorff space X into a locally convex space E, for which f(X) is relatively compact. As it is shown in 181, the uniform dual Crc of Crc can be identified with a space M(B, E') of E'-valued measures defined on the algebra of subsets of X generated by the zero sets. In this paper the subspaces of all a-additive and all 7-additive members of M(B, E') are studied. Two locally convex topologies 3 and P1 are considered on Crc. They yield as dual spaces the spaces of all T-additive and all a-additive members of M(B, E') respectively. In case E is a locally convex lattice, the a-additive and x-additive members of M(B, E') correspond to the a-additive and x-additive members of Crc respectively. 1. Definitions and preliminaries. Let X be a completely Hausdorff space and let E be a real locally convex Hausdorff space. Let Cb = Cb(X) denote the space of all bounded continuous real-valued functions on X. We will denote by Crc = Crc(X, E) the space of all continuous functions f, from X into E, for which 1(X) is relatively compact. Clearly Crc consists of those continuous functions f, from X into E, that have continuous extensions fto all of the Stone-Cech compactification ,BX of X. For an f in Cb we will denote also by fits unique continuous extension to all of ,BX. The zero sets in X are defined to be the kernels of real continuous functions on X. The complement of a zero set is called a cozero set. Let z be an algebra of subsets of X and let m be a finitely-additive bounded real set function on S. We say that m is regular with respect to a subfamily 2 of z if the following condition is satisfied: For every F in z and every e > 0 there exists G in S1 such that G C F and Im(H)I < e for all H in z which are contained in F G. Received by the editors September 24, 1972 and, in revised form, March 8, 1974. AMS (MOS) subject classifications (1970). Primary 46E40; Secondary 46A05, 46A40.