Measurability properties of real-valued functions

Abstract

In this chapter, two concepts associated with measurability properties of various real-valued functions are introduced and examined. These concepts are the notion of a relatively measurable real-valued function with respect to a given class M of measures and the notion of an absolutely nonmeasurable function with respect to M. Naturally, the usefulness of these concepts will be illustrated below by a number of relevant examples (cf. [130]). Further, a characterization of absolutely nonmeasurable functions will be established in the most important case when the role of M is played by the class M _E of all nonzero σ-finite diffused measures on a given base set E. Also, it will be shown that the functions produced by Vitali’s classical partition of the real line R are relatively measurable with respect to the class of all extensions of the Lebesgue measure λon this line. In order to obtain the latter result, Theorem 2 of Chapter 2 will be utilized.