Chapter 6 - Radon-Nikodým Theorems

Abstract

This chapter describes the various aspects of Radon–Nikodym theorems. A general Radon–Nikodym theorem for nonnegative finitely additive scalar measures is presented in the chapter from which the basic ideas of further results can be drawn. The conditions that permit even unbounded measures to have Radon–Nikodym derivatives are examined in the chapter. The results concerning Banach spaces possessing the so-called “Radon–Nikodym Property” (RNP) are presented in the chapter. The geometric concepts become essential tools for describing Banach spaces. The finitely additive measures taking values in Banach spaces and to the research of weaker types of derivatives are elaborated in the chapter. To find a characterization of the Radon–Nikodym property, the concept of completeness is required. A complex measure or function can be studied investigating separately its real and imaginary parts. It is proved in the chapter that convergence in measure implies convergence for some subsequence; hence, a strongly measurable function is essentially valuedseparably. It is found that the problems concerning the existence of a Radon–Nikodym derivative are harder when one allows also finitely additive measures into consideration. Some results concerning Radon–Nikodym derivatives for Sugeno integral are also presented in the chapter.