Abstract
The outer regular fuzzy measure and the regular fuzzy measure are introduced. The monotone convergence theorem and the representation theorem are presented. The relation of the regular fuzzy measure and Choquet capacity are discussed. Formulas for calculation of a Choquet integral of a function on the real line are shown.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Choquet, G.: Theory of capacities. Ann. Inst. Fourier, Grenoble. 5, 131–295 (1955)
Gavriluţ, A.C.: Regularity and autocontinuity of set multifunctions. Fuzzy Sets and Systems (in Press)
Denneberg, D.: Non additive measure and Integral. Kluwer Academic Publishers, Dordorecht (1994)
Dellacherie, C.: Quelques commentaires sur les prolongements de capacités. In: Séminaire de Probabilités 1969/1970, Strasbourg. Lecture Notes in Mathematics, vol. 191, pp. 77–81. Springer, Heidelberg (1971)
Grabisch, M., Nguyen, H.T., Walker, E.A.: Fundamentals of uncertainty calculi with applications to fuzzy inference. Kluwer Academic Publishers, Dordorecht (1995)
Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals: Theory and Applications. Springer, Berlin (2000)
Ji, A.B.: Fuzzy measure on locally compact space. The Journal of Fuzzy Mathematics 5(4), 989–995 (1997)
Kawabe, J.: Regularities of Riesz space-valued non-additive measures with applications to convergence theorems for Choquet integrals. Fuzzy Sets and Systems (in Press)
Masiha, H.P., Rahmati, M.: A symmetric conjugate condition for the representation of comonotonically additive and monotone functionals. Fuzzy Sets and Systems 159, 661–669 (2008)
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)
Murofushi, T., Sugeno, M.: A Theory of Fuzzy Measures: Representations, the Choquet integral and null sets. J. Math. Anal. Appl. 159, 532–549 (1991)
Narukawa, Y., Murofushi, T., Sugeno, M.: The comonotonically additive functional on the class of continuous functions with compact support. In: Proc. FUZZ-IEEE 1997, pp. 845–852 (1997)
Narukawa, Y., Murofushi, T., Sugeno, M.: Representation of Comonotonically Additive Functional by Choquet Integral. In: Proc. IPMU 1998, pp. 1569–1576 (1998)
Narukawa, Y., Murofushi, T., Sugeno, M.: Regular fuzzy measure and representation of comonotonically additive functionals. Fuzzy Sets and Systems 112, 177–186 (2000)
Narukawa, Y., Murofushi, T., Sugeno, M.: Boundedness and Symmetry of Comonotonically Additive Functionals. Fuzzy Sets and Systems 118, 539–545 (2001)
Narukawa, Y., Murofushi, T., Sugeno, M.: Extension and representation of comonotonically additive functionals. Fuzzy Sets and Systems 121, 217–226 (2001)
Narukawa, Y., Murofushi, T.: Conditions for Choquet integral representation of the comonotonically additive and monotone functional. Journal of Mathematical Analysis and Applications 282, 201–211 (2003)
Narukawa, Y., Murofushi, T.: Regular non-additive measure and Choquet integral. Fuzzy Sets and Systems 143, 487–492 (2004)
Narukawa, Y., Murofushi, T.: Choquet integral with respect to a regular non-additive measures. In: Proc. 2004 IEEE Int. Conf. Fuzzy Systems (FUZZ-IEEE 2004), pp. 517–521 (2004) (paper# 0088-1199)
Narukawa, Y.: Inner and outer representation by Choquet integral. Fuzzy Sets and Systems 158, 963–972 (2007)
Pap, E.: Regular null additive monotone set functions. Univ. u Novom Sadu Zb. rad. Prorod.-Mat. Fak. Ser. mat. 25(2), 93–101 (1995)
Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dordorechet (1995)
Pap, E., Mihailovic, B.P.: A representation of a comonotone–additive and monotone functional by two Sugeno integrals. Fuzzy Sets and Systems 155, 77–88 (2005)
Song, J., Li, J.: Regularity of null additive fuzzy measure on metric spaces. International Journal of General Systems 32(3), 271–279 (2003)
Rebille, Y.: Sequentially continuous non-monotonic Choquet integrals. Fuzzy Sets and Systems 153, 79–94 (2005)
Rebille, Y.: A Yosida-Hewitt decomposition for totally monotone set functions on locally compact σ− compact topological spaces. International Journal of Approximate Reasoning 48, 676–685 (2008)
Šipoš, J.: Non linear integral. Math. Slovaca 29(3), 257–270 (1979)
Sugeno, M.: Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology (1974)
Sugeno, M., Narukawa, Y., Murofushi, T.: Choquet integral and fuzzy measures on locally compact space. Fuzzy sets and Systems 99(2), 205–211 (1998)
Torra, V., Narukawa, Y.: Modeling decisions: Information fusion and aggregation operators. Springer, Berlin (2006)
Wu, J., Wu, C.: Fuzzy regular measures on topological spaces. Fuzzy sets and Systems 119, 529–533 (2001)
Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. on Systems, Man and Cybernetics 18, 183–190 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Narukawa, Y., Torra, V. (2010). Choquet Integral on Locally Compact Space: A Survey. In: Huynh, VN., Nakamori, Y., Lawry, J., Inuiguchi, M. (eds) Integrated Uncertainty Management and Applications. Advances in Intelligent and Soft Computing, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11960-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-11960-6_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11959-0
Online ISBN: 978-3-642-11960-6
eBook Packages: EngineeringEngineering (R0)