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Integral Sums and Integrals

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Non-Additive Measures

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

Abstract

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.

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References

  1. Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313–333 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953/1954)

    Google Scholar 

  4. Dunford, N., Schwarz, J.T.: Linear operators. Part 1. General Theory. Interscience Publ., New York (1966)

    MATH  Google Scholar 

  5. Durante, F., Sempi, C.: Semicopulæ. Kybernetika 43(2), 209–220 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Even, Y., Lehrer, E.: Decomposition-Integral: Unifying Choquet and the Concave Integrals, http://www.math.tau.ac.il/~lehrer/Papers/decomposition.pdf (submitted)

  7. Klement, E.P., Mesiar, R.: Discrete Integrals and Axiomatically Defined Functionals. Axioms 1(1), 9–20 (2012)

    Article  Google Scholar 

  8. Klement, E.P., Mesiar, R., Pap, E.: Measure-based aggregation operators. Fuzzy Sets and Systems 142(1), 3–14 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  10. Lebesgue, H.: Intègrale, longueur, aire. Habilitation thesis, Université de Paris, Paris (1902)

    Google Scholar 

  11. Lehrer, E.: A new integral for capacities. Economic Theory 39, 157–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lehrer, E., Teper, R.: The concave integral over large spaces. Fuzzy Sets and Systems 15, 2130–2144 (2008)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Mesiar, R., Li, J., Pap, E.: Discrete pseudo-integrals. International Journal of Approximate Reasoning 54(3), 357–364 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mesiar, R., Rybárik, J.: Pan-operations structure. Fuzzy Sets and Systems 74(3), 365–369 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mesiar, R.: Stupňanová, A.: Decomposition integrals. Int. Journal of Approximate Reasoning (in print, 2013)

    Google Scholar 

  17. Mesiar, R., Vivona, D.: Two-step integral with respect to fuzzy measure. Tatra Mt. Math. Publ. 16(pt. II), 359–368 (1999)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Narukawa, Y., Torra, V.: Twofold integral and Multi-step Choquet integral. Kybernetika 40(1), 39–50 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Lecture Notes in Statistics, vol. 139. Springer, New York (2006)

    MATH  Google Scholar 

  22. Pap, E.: Handbook of Measure Theory, Part I, Part II. Elsevier, Amsterdam (2002)

    Google Scholar 

  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. Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy-integral. I Kybernetika 41(3), 397–422 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy-integral. II Kybernetika 41(4), 469–496 (2005)

    MathSciNet  MATH  Google Scholar 

  26. Sander, W., Siedekum, J.: Multiplication, distributivity and fuzzy-integral. III Kybernetika 41(3), 497–518 (2005)

    MATH  Google Scholar 

  27. Schmeidler, D.: Integral representation without additivity. Proc. Amer. Math. 97(2), 255–270 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)

    Article  MathSciNet  Google Scholar 

  29. Šipoš, J.: Integral with respect to a pre-measure. Math. Slovaca 29(2), 141–153 (1979)

    Google Scholar 

  30. Sugeno, M.: Theory of Fuzzy Integrals and its Applications. PhD thesis, Tokyo Institute of Technology (1974)

    Google Scholar 

  31. Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  33. Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. In: Cognitive Technologies. Springer (2007)

    Google Scholar 

  34. Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, Heidelberg (2009)

    Book  MATH  Google Scholar 

  35. Yang, Q.: The Pan-integral on the Fuzzy Measure Space. Fuzzy Mathematics 3, 107–114 (1985)

    Google Scholar 

  36. Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Applic. 23, 421–427 (1968)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar & Andrea Stupňanová

Authors
  1. Radko Mesiar
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  2. Andrea Stupňanová
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Corresponding author

Correspondence to Radko Mesiar .

Editor information

Editors and Affiliations

  1. IIIA-CSIC, Bellaterra, Spain

    Vicenc Torra

  2. Toho Gakuen, Kunitachi, Tokyo, Japan

    Yasuo Narukawa

  3. European Centre for Soft Computing, Mieres, Spain

    Michio Sugeno

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Mesiar, R., Stupňanová, A. (2014). Integral Sums and Integrals. In: Torra, V., Narukawa, Y., Sugeno, M. (eds) Non-Additive Measures. Studies in Fuzziness and Soft Computing, vol 310. Springer, Cham. https://doi.org/10.1007/978-3-319-03155-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-03155-2_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03154-5

  • Online ISBN: 978-3-319-03155-2

  • eBook Packages: EngineeringEngineering (R0)

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