Continuity properties of vector measures

Abstract

This paper is a sequel to [ I 1. Using projections in the universal measure space of W. H. Graves [ 6 1, we study continuity and orthogonality properties of vector measures. To each strongly countably additive measure 4 on an algebra .d with range in a complete locally convex space we associate a projection P,, which may be thought of as the support of @ and as a projection-valued measure. Our main result is that, as measures, # and P, are mutually continuous. It follows that measures may be compared by comparing their projections. The projection P, induces a decomposition of vector measures which turns out to be Traynor’s general Lebesgue decomposition [8) relative to qs. Those measures $ for which algebraic d-continuity is equivalent to $continuity are characterized in terms of projections, while $-continuity is shown to be equivalent to &order-continuity whenever -d is a u-algebra. Using projections associated with nonnegative measures, we approximate every 4 uniformly on the algebra ,d by vector measures which have control measures. Last, we interpret our results algebraically. Under the equivalence relation of mutual continuity, and with the order induced by continuity, the collection of strongly countably additive measures on .d which take values in complete spaces forms a complete Boolean algebra 1 rV, which is isomorphic to the Boolean algebra of projections in the universal measure space. The homomorphic image of the lattice of nonnegative measures on .d is a dense u-ideal in 4. Using the results in this paper, the author and W. H. Graves have investigated the range of a vector measure [ 3 1 and characterized the closed measures of Kluvanek (21. All definitions and notation are as in [ 11. In particular, .d is a fixed algebra of subsets of a nonempty set X and L(.d) is the universal measure 268 0022-247X/82/030268-13$02.00/O