Measures on Locally Compact Spaces

This final chapter of the book is concerned with topological measure theory and integral representation. The domain X of the operator-valued measure θ will be a locally compact topological space and the σ-ring $\Re$ will consist of all relatively compact Borel subsets of X. As usual, measures on topological spaces are supposed to fulfill certain regularity requirements. Sections 1 and 2 of this chapter will probe different notions of continuity for locally convex cone-valued functions on a topological space. Inductive limit-type topologies lead to the identification of certain cones of continuous functions. This is motivated by the concept of weighted spaces of continuous real-valued functions which is due to Nachbin and Prolla. Continuous linear operators on such cones of cone-valued functions will be investigated in Section 4. The main result is a generalized Riesz-type integral representation theorem for this type of operators in Section 5. Section 6 contains a long list of special cases and examples, including a generalization of the classical Spectral representation theorem for normal linear operators on a complex Hilbert space.

Generally, the notations introduced in Chapters I and II will be used. As usual, for a subset Y of a topological space X, the sets $\overline {Y},\,Y^\circ$ and $\partial Y = \overline {Y}\backslash Y^\circ$ denote its topological closure, interior and boundary in X, respectively. The core support of a cone-valued function f on X is the set $\{ x \in X\ |\ f(x) \ne 0\}$ and denoted by supp*(f). Its closure, supp(f) is the usual support of f.