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Zbl 0972.28010
Butnariu, Dan; Klement, Erich Peter
Measures on triangular norm-based tribes: Properties and integral representations.
(English)
[A] Grabisch, Michel (ed.) et al., Fuzzy measures and integrals. Theory and applications. Heidelberg: Physica-Verlag. Stud. Fuzziness Soft Comput. 40, 233-246 (2000). ISBN 3-7908-1258-7/hbk

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 {\it G. Barbieri} and {\it H. Weber} [J. Math. Anal. Appl. 244, No. 2, 408-424 (2000; Zbl 0965.28010)].
[Hans Weber (Udine)]
MSC 2000:
*28E10 Fuzzy measures

Keywords: Lyapunov theorem; triangular norm-based tribes; integral representations; Jordan decomposition

Citations: Zbl 0965.28010

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