## 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)

### MSC:

 2.8e+11 Fuzzy measure theory

Zbl 0965.28010