Measures and Integrals. The General Theory

In this chapter we shall develop a general integration theory for cone-valued functions with respect to operator-valued measures. The structure of locally convex cones will allow the use of many of the main concepts of classical measure theory for (extended) real-valued functions. Section 1 introduces measurability for cone-valued functions on a set X with respect to a (weak) σ-ring of subsets of X. This notion does not involve any reference to a particular measure. Bounded operator-valued measures will be defined in Section 3. The introduction of its modulus allows the extension of any given measure to a full locally convex cone containing the given cone and its neighborhood system, thus greatly facilitating the expansion of our concepts. This yields a new understanding of the variation of a measure, not as a separate positive real-valued measure associated with the given one, but as a component of its extension. The development of an integration theory for cone-valued functions with respect to an operator-valued measure follows in Section 4. Section 5 contains the general convergence theorems for sequences of functions and measures, that is variations and adaptations of the dominated convergence theorem. Chapter II concludes with a long list of special cases and examples in Section 6, demonstrating the generality of the approach. These examples include classical real-valued measure theory as well as settings with vector-,cone-, functional- and operator-valued measures and functions.