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)].