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Max-min (σ-)additiye representation of monotone measures

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Abstract

In non-additive measure and integration (or fuzzy measure and integral) one tries to generalise the issues of product measure and conditional expectation from the additive theory. In the discrete case successful attempts have been made via the max-min additive representation of the monotone measure and the corresponding integrals.

The present paper intends to find, for arbitrary monotone measures, a maxmin additive representation and, under certain topological assumptions, a representation with σ-additive measures, thus providing a powerful tool for the theory of non-additive measure and integration.

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References

  1. F.F. Bonsall: Sublinear functionals and ideals in partially ordered vector spaces. Proc. London Math. Soc. 3 (1954), 402–418.

    Article  MathSciNet  Google Scholar 

  2. A. Chateauneuf: Combination of compatible belief functions and relation of specifity, p. 97–114 in M. Fedrizzi, J. Kaprczyk and R.R. Yager: Advances in the Dempster-Shafer Theory of Evidence, Wiley, 1994.

  3. G. Choquet: Forme abstraite du Théorème de capacitabilite. Annales de 1’Institut Fourier 9 (1959), 83–89.

    MATH  MathSciNet  Google Scholar 

  4. F. Delbaen: Convex games and extreme points. J. Math Analysis Appl. 45 (1974), 210–233.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Dellacherie, P. Meyer: Probabilities and Potential. North Holland Publishing Company, Amsterdam 1978.

    MATH  Google Scholar 

  6. D. Denneberg: Non-additive Measure and Integral. Theory and Decision Library: Series B, Vol 27. Kluwer Academic, Dordrecht, Boston 1994, 2. ed. 1997.

    Google Scholar 

  7. D. Denneberg: Totally monotone core and products of monotone measures. International Journal of Approximate Reasoning 24 (2000), 273–281.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Denneberg: Conditional expectation for monotone measures, the discrete case. Preprint (www.informatik.uni-bremen.de/~denneberg), Universitat Bremen 2000.

  9. D. Denneberg and M. Grabisch: Interaction Transform of Set Functions over a Finite Set. Information Sciences 121 (1999), 149–170.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Parker: The sigma-core of a cooperative game. Manuscripta Mathematica 70 (1991), 247–253.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Schmeidler: Cores of exact games I. J. Math. Ananlysis. Appl. 40 (1972), 214–225.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. Topsøe: Compactness in spaces of measures. Studia Mathematika 36 (1970), 195–212

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich 3, Universität Bremen, 28334, Bremen, Germany

    Martin Brüning & Dieter Denneberg

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  1. Martin Brüning
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  2. Dieter Denneberg
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Correspondence to Martin Brüning or Dieter Denneberg.

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Brüning, M., Denneberg, D. Max-min (σ-)additiye representation of monotone measures. Statistical Papers 43, 23–35 (2002). https://doi.org/10.1007/s00362-001-0084-5

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  • DOI: https://doi.org/10.1007/s00362-001-0084-5

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