The Vitali-Hahn-Saks and Nikodym theorems for additive set functions
Richard B. Darst · Bulletin of the American Mathematical Society · 1970
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 {ju n } 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) =lim w ix n {E) exists f or every ££2, then \x is bounded and additive and the additivity of the /x w is uniform in n.A MS 1969 subject classifications.Primary 2813,2850.