# zbMATH — the first resource for mathematics

Choquet-like integrals with respect to level-dependent capacities and $$\varphi$$-ordinal sums of aggregation function. (English) Zbl 1340.28020
Summary: In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a $$\varphi$$-ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regarded as extension method. The approach is illustrated by several examples.

##### MSC:
 2.8e+11 Fuzzy measure theory 2.8e+06 Nonstandard measure theory
Full Text:
##### References:
 [1] Choquet, G.: Theory of capacities. Annales de l’Institite Fourier 5 (1953-1954), 131-295. · Zbl 0679.01011 [2] Denneberg, D.: Non-additive Measure and Integral. Theory and Decision Library. Series B: Mathematical and Statistical Methods 27. Kluwer Academic Publishers Group, Dordrecht 1994. · Zbl 0968.28009 [3] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge 2009. · Zbl 1206.68299 [4] Greco, S., Matarazzo, B., Giove, S.: The Choquet integral with respect to a level dependent capacity. Fuzzy Sets and Systems 175 (2011), 1-35. · Zbl 1218.28014 [5] Mesiar, R.: Choquet-like integrals. J. Math. Anal. Appl. 194 (1995), 477-488. · Zbl 0845.28010 [6] Mesiar, R., Baets, B. De: New construction methods for aggregation operators. Proc. IPMU’2000, Madrid, pp. 701-706. [7] Pap, E.: Handbook of Measure Theory. Vol. I, II. North-Holland, Amsterdam 2002. · Zbl 0998.28001 [8] Pap, E.: An integral generated by a decomposable measure. Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20 (1990), 135-144. · Zbl 0754.28002 [9] Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy integral, I, II, III. Kybernetika 41 (2005) I: 397-422, II: 469-496, III: 497-518. · Zbl 1249.28028 [10] Schmeidler, D.: Integral representation without additivity. Proc. Amer. Math. Soc. 97 (1986), 255-261. · Zbl 0687.28008 [11] Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57 (1989), 571-87. · Zbl 0672.90011 [12] Sugeno, M.: Theory of Fuzzy Integrals and its Applications. PhD Thesis, Tokyo Institute of Technology, 1974. [13] Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122 (1987), 197-222. · Zbl 0611.28010 [14] Vitali, G.: Sula definizione di integrale delle funzioni di una variabile. Ann. Mat. Pura Appl. 2 (1925), 111-121. · JFM 51.0201.01 [15] Wang, Z., Klir, G. J.: Generalized Measure Theory. Springer, 2009. · Zbl 1184.28002
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.