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The purpose of this note is twofold. Firstly, we point out an appropriate version of the Vitali-Hahn-Saks and Nikodym theorems for finitely additive set functions defined on a sigma algebra of sets. Secondly, we apply our Theorem to extend a recent result of James K. Brooks for countably additive vector valued set functions to the general finitely additive case. THEOREM. Let {jun} be a sequence of bounded and finitely additive scalar valued set functions defined on a sigma algebra 2 of subsets of a set S. If fx(E) =limw ixn{E) exists f or every ££2, then \x is bounded and additive and the additivity of the /xw is uniform in n. In addition, suppose lim^^o jun(£)=0 for each n, where v is a nonnegative finitely additive set function defined on 2. Then limV(jE?)^oMn(£) =0 uniformly in n. Our Theorem follows from the weak convergence theory for finitely additive set functions. While a proof of it can be synthesized (modulo an observation) from (3), (5), (6) and (7), for the readers' con­ venience we shall also refer to the discussion of the work of Soloman Leader and Pasquale Porcelli (ó) and (7) given in (4). The corollary on p. 475 of (3) tells us that the sequence {JUW} is weakly convergent. Also, the (L)-space of bounded and additive functions on 2 is weakly complete (5, Theorem 12), so the sequence {fJLn} converges weakly to li. Hence JU is bounded and additive, and (cf. (4)) the weakly con­ vergent sequence {/Xn} is equi-absolutely continuous with respect to the bounded and additive function <p defined on 2 by (1) <P(E) = £ 2-*(l + I M» I (S))"11 M» I CE),

The Vitali-Hahn-Saks and Nikodym theorems for additive set functions