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For ,u, v E ba+(Q, A) it is shown that the existence of a net of nonnegative functions f, that are primitive relative to al and satisfy lim, fA f, du = v (A), A E 2l, is equivalent to the condition v ' ,u, i.e. ,(A) = 0 for some A E l implies v (A) = 0. Furthermore, as an application it is proved that for ,u, v E ba+(Q, A) satisfying v E , and any extension u' E ba+(Q, 2') of u, where 2t' denotes some algebra of subsets of Q containing 2l, there exists some extension v' E ba+ (Q, 2) of v such that z I g' ,u is valid. Let ba+(Q, %2) denote the subset of the set ba(Q, 2) of all bounded and finitely-additive set functions on the algebra A of subsets of the set Q that are nonnegative. Furthermore, B(Q, A) stands for the family of all real-valued functions on Q that are uniform limits of functions of the type EnZ OijA a E R, A1 E A, i = 1, ..., n, n ENN ("primitive function relative to A"). Finally, v is called weakly absolutely continuous with respect to ,u (symbol: v 0 there exists some 3 > 0 such v(A) ?0 for all Ac E for some f E B(Q, 2) implies fA f dv >0 for all A E A. This last condition is equivalent to the relation v u ,u, since one can replace f E B(Q, A) by some function f: Q -i R, which is primitive relative to A. According to a well-known uniform version of the Radon-Nikodym theorem in the finitely-additive case due to Bochner, there exists a net ft of nonnegative Received by the editors February 8, 1990 and, in revised forms, May 20, 1990 and August 15, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28A12, 28A20.

A nonuniform version of the theorem of Radon-Nikodým in the finitely additive case with applications to extensions of finitely additive set functions