Abstract. In this paper we extend the well-known Vitali-Hahn-Saks and Nikod5rm theorems for measures to finitely additive vector-valued set functions.1. Introduction. In ref. 1, Brooks gave new proofs for the Vitali-HahnSaks'0'5'9 and NikodQm6 theorems for countably additive set functions. The main tool was the Schur theorem.2 By establishing a vector form of Phillips' theorem,7 which in turn generalizes Schur's theorem, we are able to extend the Nikodym theorem (Theorem 2) and the Vitali-Hahn-Saks theorem (Theorem 3) to the finitely additive vector case.All the results in this paper, with modifications, are valid for locally convex topological vector spaces. Although for simplicity the proofs are given for Banach spaces, the techniques can be adapted to apply to the more general case.2. Preliminaries. z is a v-ring of subsets of a set S. ,P(%) is the power set of the set of natural numbers M. C denotes the complex number field. E and A are generic notations for sets belonging to 2 and 6P(9Z) respectively.

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