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Throughout this paper, let X be a completely regular Hausdorff space, Cb(X) the space of all real-valued bounded and continuous functions on X with supremum norm, M(X) the set of all positive linear functional // on C6(X) such that /*(!) = 1, (M(X) is also the set of all positive, regular finitely additive measures, each of total mass 1, on the algebra generated by zero sets.)* A a subspace of Cb(X) which separates points of X and contains the constant functions, and A* the set of all real-valued continuous linear functionals on A. If X is a non-empty closed, bounded and convex subset of an LCHTVS (locally convex Hausdorff topological vector space) E over the field -R of real numbers, then we always regard A as the subspace {/|x+r :/<E£*, r<=R} of C b ( X ) , where E* is the topological dual of E, and/U is the restriction of / to X. Denote by K(A) the set of all L in A* such that L(l) = 1 = ||L||. If we consider A* (resp. Cj(X)*) in its weak* topology, then K(A) (resp. M(X)) is a non-empty compact convex subset of an LCHTVS A* (resp. Cb(X) *) over R. If a is in X, let 0(a) be the element of K(A) defined by (j) (a) (/) =/(a) for any / in A, and s(a) the element of M(X) defined by e(<z) (/) =/(a) for any / in C6(X). Note that ^ is a oneto-one and continuous mapping from X into K(A). The purpose of this paper is to give a characterization of point a in X with the following property (*).

On general denting points and the unique positive extension of certain positive linear functionals