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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
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References:

[1] Denneberg, D., Non-Additive Measure and Integral (1997), Kluwer: 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.; M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology, 1974.
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