Measures on triangular norm-based tribes: Properties and integral representations. (English) Zbl 0972.28010

Grabisch, Michel (ed.) et al., Fuzzy measures and integrals. Theory and applications. Heidelberg: Physica-Verlag. Stud. Fuzziness Soft Comput. 40, 233-246 (2000).
The paper gives an overview about real-valued measures on triangular norm-based tribes. The domain of these measures are certain classes of fuzzy sets (i.e., \([0,1]\)-valued functions) which are proper generalizations of \(\sigma\)-algebras and where the set theoretical operations are derived from triangular norms. The authors first present several particular triangular norms \(T\) and discuss the corresponding \(T\)-tribes. For measures on \(T\)-tribes with respect to particular triangular norms \(T\), the paper contains integral representations, a Jordan decomposition theorem and a Lyapunov type theorem. The paper concludes with several open problems some of which are related to the article of G. Barbieri and H. Weber [J. Math. Anal. Appl. 244, No. 2, 408-424 (2000; Zbl 0965.28010)].
For the entire collection see [Zbl 0935.00014].
Reviewer: Hans Weber (Udine)


28E10 Fuzzy measure theory


Zbl 0965.28010