On The Vitali-Hahn-Saks Theorem

In this paper we generalize the classical Vitali-Hahn-Saks theorem to sets of countably additive vector measures which are compact in the strong operator topology. The main result asserts that a set of countably additive vector measures which is compact in the strong operator topology is uniformly countably additive. We accomplish this by first studying the properties of linear operators from Y*, the dual of a Banach space Y, into a Banach space X which are continuous with respect to the Mackey topology τ(Y*,Y) on Y* and the norm topology on X, and then applying the results to the special case where Y = L1(μ) and Y* = L∞(μ). Other related results on vector measures are also included.