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Some quantities represented by the Choquet integral. (English) Zbl 0816.28012
The authors show how one can use the Choquet integral to express several quantities in probability theory and statistics. Those quantities include, among others, supremum, infimum, essential supremum, essential infimum, mean, median and \(\alpha\)-quantile.
Reviewer: T.Kubiak (Poznań)

28E10 Fuzzy measure theory
Full Text: DOI
[1] Bannon, G., Distinction between several subsets of fuzzy measures, Fuzzy sets and systems, 5, 291-305, (1981) · Zbl 0449.60001
[2] Choquet, G., Theory of capacities, Ann. inst. Fourier, 5, 131-295, (1953) · Zbl 0064.35101
[3] de Campos, L.M.; Lamata, M.T.; Moral, S., A unified approach to define fuzzy integrals, Fuzzy sets and systems, 39, 75-90, (1991) · Zbl 0773.28015
[4] Delgado, M.; Moral, S., Upper and lower fuzzy measures, Fuzzy sets and systems, 33, 191-200, (1989) · Zbl 0677.28009
[5] Dempster, A.P., Upper and lower probabilities induced by a multivalued mapping, Ann. math. statist., 38, 325-339, (1967) · Zbl 0168.17501
[6] Murofushi, T.; Sugeno, M., An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy sets and systems, 29, 201-227, (1989) · Zbl 0662.28015
[7] Murofushi, T.; Sugeno, M., A theory of fuzzy measures: representations, the Choquet integral, and null sets, J. math. anal. appl., 159, 532-549, (1991) · Zbl 0735.28015
[8] Shafer, G., A mathematical theory of evidence, (1976), Princeton University Press Princeton, NJ · Zbl 0359.62002
[9] Smets, Ph., The degree of belief in a fuzzy event, Inform. sci., 25, 1-19, (1981) · Zbl 0472.62005
[10] Sugeno, M., Theory of fuzzy integrals and its applications, Doctoral thesis, Tokyo institute of technology, (1974)
[11] Tsukamoto, Y., A measure theoretic approach to evaluation of fuzzy set defined on probability space, Fuzzy math., 2, 3, 89-98, (1982)
[12] Wakker, P., A behavioral foundation for fuzzy measures, Fuzzy sets and systems, 37, 327-350, (1990) · Zbl 0716.90023
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