The order dual of the space of Radon measures

Abstract

Introduction. In [14] and [15] S. Kaplan studied the second dual of the Banach lattice of all continuous real-valued functions on a compact space. Then in [16] he initiated a study of the second dual of the lattice of continuous functions with compact support on a locally compact space. It is the purpose of this paper to continue the study of the locally compact case. For a locally compact space X, let Lk denote the vector lattice of Radon measures on X. In ?2 the basic properties of Lk are established. In ?3 we devote our attention to the proof of the following theorem: Every purely nonatomic measure defined on a c-compact space can be extended to a measure on a countably compact space. Given a compact subset K of X, let L(K) denote the set of Radon measures on K. Then L(K) can be identified with an ideal in Lk. The set UL(K), where the union is taken over all compact subsets of X, is the set of all measures with compact support. It appears that the order dual M of UL(K) is an appropriate object of study as well as the order dual Mk of Lk and the order dual Mb of the space Lb of all finite measures. In particular, C (the set of all continuous realvalued functions on X) can be embedded in M while in general this is not possible for either Mk or Mb. The spaces Mk and Mb appear as ideals in M. Also, M can be characterized as the set of all multiplication operators on Lk. In ?5 we consider the question of whether Mk can be identified with a set of continuous functions with compact support. This is the question raised by Kaplan in [16, ??5,7]. After giving an example which shows that in general this is not possible, we state conditions which are sufficient to insure that Mk will be the set of continuous functions with compact support on some locally compact space. In ?6 we turn our attention to the duality relations which exist between the ideals in Lk and those in M. It will be assumed that the reader is familiar with Kaplan's papers, On the second dual of the space of continuousfunctions [14], [15], [16].