Chapter 26 - Positive and Complex Radon Measures in Locally Compact Hausdorff Spaces

Abstract

This chapter discusses positive and complex radon measures in locally compact Hausdorff spaces. Positive measures mean countably additive nonnegative valued set functions defined on a ring of sets. A boundedly complete vector lattice under the partial ordering is given in the chapter. Several notions of regularity for measures defined on certain rings or σ-rings of subsets and that study the existence and uniqueness of regular extensions are presented in the chapter. The complex Radon measures and their properties are explained in the chapter. Using the various notions of regularity for positive and complex measures, the regular extensions of positive and complex measures are studied. Several characterizations of a bounded linear functional are elaborated in the chapter. The characterization of the complex Radon measures in terms of complex measures is also described in the chapter. The vector space of all complex valued additive set functions of finite variation on a ring of sets is shown in the chapter to be isomorphic