Chapter 22 - Measures on Clans and on MV-Algebras

Abstract

This chapter discusses the measures on clans and on MV-algebras. The results of certain type of fuzzy measures valuations on clans of fuzzy sets are presented in the chapter. A basic example of an MV-algebra is an order interval of a commutative group. The center of MV algebra has a basic role in decomposition theorems and in the representation of measures. The Loomis–Sikorski theorem for Boolean algebras says that every σ -complete Boolean algebra is isomorphic to σ-complete Boolean algebra of point sets modulo, an σ-ideal in that algebra. Any complete Boolean algebra is isomorphic to the product of an atomless Boolean algebra. This result will be applied to obtain a topological decomposition of uniform MV-algebras, which is a basic tool to prove the decomposition theorem. The Jordan decomposition theorem valid in any Riesz space yields the decomposition theorem for measures. The Hahn decomposition theorem for measures on MV-algebras can be derived from the Jordan decomposition theorem. It is shown in the chapter that Butnariu's integral representation theorem is an easy consequence of the Hahn decomposition theorem.