An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure. (English) Zbl 0662.28015

In this paper the non-additivity of Sugeno’s fuzzy measure is interpreted in terms of addition and the rationality of the Choquet integral is discussed. It is pointed out that a fuzzy measure on a set X expresses the interaction between the subsets of X and can be represented by an additive measure. It is shown through concrete examples that the Choquet integral is reasonable as an integral with respect to a fuzzy measure. It is also found that the Choquet integral is closely related with the representation of a fuzzy measure.


28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Choquet, G., Theory of capacities, Ann. inst. Fourier, 5, 131-295, (1953) · Zbl 0064.35101
[2] Dempster, A.P., Upper and lower probabilities induced by a multivalued mapping, Ann. math. statist., 38, 325-339, (1967) · Zbl 0168.17501
[3] Höhle, U., A mathematical theory of uncertainty, (), 344-355
[4] Höhle, U., Integration with respect to fuzzy measures, (), 35-37
[5] Ishii, K.; Sugeno, M., A model of human evaluation process using fuzzy measure, Internat. J. man-machine stud., 22, 19-38, (1985) · Zbl 0567.90059
[6] Krantz, D.H.; Luce, R.D.; Suppes, P.; Tversky, A., ()
[7] Kruse, R., Fuzzy integrals and conditional fuzzy measures, Fuzzy sets and systems, 10, 309-313, (1983) · Zbl 0525.28001
[8] Ralescu, D., Toward a general theory of fuzzy variables, J. math. anal. appl., 86, 176-193, (1982) · Zbl 0482.60004
[9] Seif, A.; Aguilar-Martin, J., Multi-group classification using fuzzy correlation, Fuzzy sets and systems, 3, 109-122, (1980) · Zbl 0426.68084
[10] Sugeno, M., Theory of fuzzy integrals and its applications, () · Zbl 0316.60005
[11] Sugeno, M.; Murofoshi, T., Pseudo-additive measures and integrals, J. math. anal. appl., 122, 197-200, (1987) · Zbl 0611.28010
[12] Weber, S., ⊥-decomposable measures and integrals for Archimedean t-conorms ⊥, J. math. anal. appl., 101, 114-138, (1984) · Zbl 0614.28019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.