13 - Triangular norms and measures of fuzzy sets

Abstract

The classical measure theory is a basis of many mathematical disciplines, including integration and probability. In contrast to it, fuzzy set theory became successful in description of systems with vague information and graded truth values. To see similarities and differences of the classical and fuzzy measure theory, the classical definition of a charge and other necessary notions used in the ordinary measure and probability theory is recalled. For simplicity, restriction is made to the real-valued measures which are finite. On the other hand, the discussion is mostly with charges as this approach appears natural in the generalization of measures to collections of fuzzy sets. The alternate definitions mentioned in this chapter forms the basis of generalizations when passing to measures and charges on collections of fuzzy sets. In a very general context, the fuzzy measure theory is sometimes used to denote any non-additive extension of the classical measure theory. The underlying collection of fuzzy sets has to satisfy some requirements to make the definition of measure meaningful.