ABSTRACT

Let E be a separable locally compact Hausdorff space. In this section we in-

troduce the space M(E), consisting of Radon signed measures on B(E), the Borel subsets of E , with compact support in E , and discuss properties of linear

functionals onM(E). We begin with basic definitions. Definition 7.1.1. Let E be a locally compact Hausdorff second-countable

topological space (i.e., with a countable base for its topology). The support

of a signed measure µ on E, denoted by supp(µ), is defined to be the complement of the largest open set O ⊆ E, such that ∣µ ∣(O) = 0, where ∣µ ∣ is the total variation measure of µ , that is, ∣µ ∣ = µ+ +µ−, where µ+ and µ− are unique mutually singular measures on E, such that µ = µ+ −µ− (this decomposition of µ is called the Jordan decomposition, [110]). We define M(E) as the set of Radon signed measures with compact support in E.