On the Existence of a Control Measure for Strongly Bounded Vector Measures

In this note we extend a theorem of Bartle, Dunford, and Schwartz [ l ] which states that for every countably additive measure denned on a (T-algebra there exists a positive "control measure" v such that v{E)—>0 if and only if ||/z||(.E) —»0, where ||/x|| is the semivariation of JJL. In this paper, ju, which is defined on a ring S, is assumed to be finitely additive and strongly bounded (^-bounded) [8] (that is n(Ei)-+0 whenever {E t} is a disjoint sequence of sets). The existing decomposition and extension theorems for vector measures can now be easily deduced by using the control measure. These applications will be presented in [2], 36 is a Banach space over the reals (the complex case is treated in a similar fashion); S* is the unit sphere in the conjugate space of 36. cr(g) denotes the cr-algebra generated by the class of sets 8. A ô-ring is a ring of sets closed under countable intersections.