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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., $\left[0,1\right]$-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)].
##### MSC:
 2.8e+11 Fuzzy measure theory