Elements of Measure Theory and Extension Constructions

Abstract

In the previous chapter a topological basis of extension was given. But realization of the general idea above requires constructing the space of GE. It is useful to note that GE are very often defined as measures. In this chapter we also follow this approach. We consider the spaces of measures and employ these spaces as model TS (see Section 3.4). Of course, we use properties of measures selectively. Firstly, we mean questions concerning the identification of measures and linear functionals. In this connection we note the well known Riesz theorem on representation of linear functionals on the space of continuous functions. In some cases it is required to consider such functionals on the space of discontinuous functions. For a respective representation we need to use finitely additive measures (FAM). On this basis the theory of extension for problems with integral constraints was suggested in [32, 35, 45, 46]. Constructions based on the Riesz theorem are used for the extension of control problems with geometric constraints (see, for example, [78, 117, 120]). In this chapter we discuss in detail constructions connected with integration with respect to FAM. In particular, we consider questions connected with the representation of linear continuous functionals on a space of discontinuous functions. This representation is very important for constructing compactifications which are used both in constructions similar to those considered in the previous chapter and in some constructions connected with problems of functional analysis. In particular, we touch upon some questions connected with the problem of universal integrability of bounded functions (see, for example, [3, 22, 92]). We use (0, 1)-measures as ‘the material’ for compactifications.