## A note on upper and lower Sugeno integrals.(English)Zbl 1094.28012

The Sugeno integral for functions measurable with respect to a paving $${\mathcal A}$$ (system of subsets of a universe $$X$$ containing the empty set) and fuzzy measures on $${\mathcal A}$$ (monotone $${\mathcal A}\to [0,1]$$ set functions vanishing at the empty set) is introduced and discussed, and extended for arbitrary $$X\to [0,1]$$ functions in two ways: as an upper and a lower Sugeno integral (compared with the standard extensions to an upper and a lower measure in the classical measure theory). Some properties are discussed and illustrated by an example.

### MSC:

 28E10 Fuzzy measure theory 06F30 Ordered topological structures

### Keywords:

fuzzy measures; Sugeno integral
Full Text:

### References:

 [1] Denneberg, D., Non-additive measure and integral, (1997), Kluwer Dordrecht [2] Kandel, A.; Byatt, W.J., Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proc. IEEE, 66, 1619-1639, (1978) [3] Murofushi, T.; Sugeno, M., A theory of fuzzy measurerepresentation, the Choquet integral and null sets, J. math. anal. appl., 159, 532-549, (1991) · Zbl 0735.28015 [4] M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology, 1974.
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