Integral Sums and Integrals

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 310)


We discuss a new approach to integration introduced recently by Even and Lehrer and its relationship to several integrals known from the literature. Decomposition integrals are based on integral sums related to some (possibly constraint) systems of set systems, such as finite chains or finite partitions. A special stress is put on the integrals which are simultaneously decomposition integrals and universal integrals in the sense of Klement et al. Several examples illustrate the presented integrals.


Choquet integral decomposition integral Sugeno integral universal integral 


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  1. 1.
    Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313–333 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benvenuti, P., Mesiar, R., Vivona, D.: Monotone set functions-based integrals. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1329–1379 (2002)Google Scholar
  3. 3.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953/1954)Google Scholar
  4. 4.
    Dunford, N., Schwarz, J.T.: Linear operators. Part 1. General Theory. Interscience Publ., New York (1966)MATHGoogle Scholar
  5. 5.
    Durante, F., Sempi, C.: Semicopulæ. Kybernetika 43(2), 209–220 (2007)MathSciNetMATHGoogle Scholar
  6. 6.
    Even, Y., Lehrer, E.: Decomposition-Integral: Unifying Choquet and the Concave Integrals, (submitted)
  7. 7.
    Klement, E.P., Mesiar, R.: Discrete Integrals and Axiomatically Defined Functionals. Axioms 1(1), 9–20 (2012)CrossRefGoogle Scholar
  8. 8.
    Klement, E.P., Mesiar, R., Pap, E.: Measure-based aggregation operators. Fuzzy Sets and Systems 142(1), 3–14 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Transactions on Fuzzy Systems 18, 178–187 (2010)CrossRefGoogle Scholar
  10. 10.
    Lebesgue, H.: Intègrale, longueur, aire. Habilitation thesis, Université de Paris, Paris (1902)Google Scholar
  11. 11.
    Lehrer, E.: A new integral for capacities. Economic Theory 39, 157–176 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lehrer, E., Teper, R.: The concave integral over large spaces. Fuzzy Sets and Systems 15, 2130–2144 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marichal, J.: An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framewor. IEEE Transactions on Fuzzy Systems 9(1), 164–172 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mesiar, R., Li, J., Pap, E.: Discrete pseudo-integrals. International Journal of Approximate Reasoning 54(3), 357–364 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mesiar, R., Rybárik, J.: Pan-operations structure. Fuzzy Sets and Systems 74(3), 365–369 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mesiar, R.: Stupňanová, A.: Decomposition integrals. Int. Journal of Approximate Reasoning (in print, 2013)Google Scholar
  17. 17.
    Mesiar, R., Vivona, D.: Two-step integral with respect to fuzzy measure. Tatra Mt. Math. Publ. 16(pt. II), 359–368 (1999)MathSciNetMATHGoogle Scholar
  18. 18.
    Mostert, P.S., Shield, A.L.: On the structure of semigroups on a compact manifold with boundary. Ann. of Math. 65, 117–143 (1957)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Murofushi, T., Sugeno, M., Fujimoto, K.: Separated hierarchical decomposition of the choquet integral. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5(5), 563–585 (1997)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Narukawa, Y., Torra, V.: Twofold integral and Multi-step Choquet integral. Kybernetika 40(1), 39–50 (2004)MathSciNetMATHGoogle Scholar
  21. 21.
    Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Lecture Notes in Statistics, vol. 139. Springer, New York (2006)MATHGoogle Scholar
  22. 22.
    Pap, E.: Handbook of Measure Theory, Part I, Part II. Elsevier, Amsterdam (2002)Google Scholar
  23. 23.
    Riemann, B.: Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe (On the representability of a function by a trigonometric series). Habilitation thesis, University of Göttingen (1854)Google Scholar
  24. 24.
    Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy-integral. I Kybernetika 41(3), 397–422 (2005)MathSciNetMATHGoogle Scholar
  25. 25.
    Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy-integral. II Kybernetika 41(4), 469–496 (2005)MathSciNetMATHGoogle Scholar
  26. 26.
    Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy-integral. III Kybernetika 41(3), 497–518 (2005)MATHGoogle Scholar
  27. 27.
    Schmeidler, D.: Integral representation without additivity. Proc. Amer. Math. 97(2), 255–270 (1986)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Šipoš, J.: Integral with respect to a pre-measure. Math. Slovaca 29(2), 141–153 (1979)Google Scholar
  30. 30.
    Sugeno, M.: Theory of Fuzzy Integrals and its Applications. PhD thesis, Tokyo Institute of Technology (1974)Google Scholar
  31. 31.
    Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Stupňanová, A.: A Note on Decomposition Integrals. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part IV. CCIS, vol. 300, pp. 542–548. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  33. 33.
    Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. In: Cognitive Technologies. Springer (2007)Google Scholar
  34. 34.
    Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, Heidelberg (2009)CrossRefMATHGoogle Scholar
  35. 35.
    Yang, Q.: The Pan-integral on the Fuzzy Measure Space. Fuzzy Mathematics 3, 107–114 (1985)Google Scholar
  36. 36.
    Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Applic. 23, 421–427 (1968)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovakia

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