Skip to main content
SpringerLink
Log in

The Monge–Kantorovich Metric on Multimeasures and Self–Similar Multimeasures

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

For a metric space \((\mathbb {X},d)\) the classical Monge-Kantorovich metric d M gives a distance between two probability measures on \(\mathbb {X}\) which is tied to the underlying distance d on \(\mathbb {X}\) in an essential way. In this paper, we extend the Monge-Kantorovich metric to signed measures and set-valued measures (multimeasures) and, in each case, prove completeness of a suitable space of these measures. Using this extension as a framework, we construct self-similar multimeasures by using an IFS-type Markov operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alo, R., de Korvin, A., Roberts C. Jr.: On some properties of continuous multimeasures. J. Math. Anal. Appl. 75, 402.410 (1980)

    Article  Google Scholar 

  2. Artstein, Z.: Set-valued measures. Trans AMS 165, 103–125 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baron, K., Lasota, A.: Markov operators on the space of vector measures; coloured fractals. Ann. Polon. Math. 69(3), 217–234 (1998)

    MATH  MathSciNet  Google Scholar 

  4. Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A 399, 243–275 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barnsley, M.F., Hutchinson, J.E., Stenflo, O.: V-variable fractals: fractals with partial self similarity. Adv. Math. 218(6), 2051–2088 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bolker, E.D.: A class of convex bodies. Trans. Amer. Math. Soc. 145, 323–345 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brooks, J.K.: An integration theory for set-valued measures. Bull. Soc. Roy. de Sci. Liege 37(5), 312–319 (1968)

    MathSciNet  Google Scholar 

  8. Debreu, G., Schmeidler, D.: The Radon-Nikodym derivative of a correspondence. In: Lecam, L., Neyman, J., Scott, E.L. (eds.) Proceedings of Fifth Berkeley Symposium Math.Stat. and Probability II, Part 1, pp 351–372. Berkely: University of California Press (1967)

  9. Drewnowski, L.: Additive and countably additive correspondences. Comment. Math. 19, 25–53 (1976)

    MATH  MathSciNet  Google Scholar 

  10. Falconer, K.J., O’Neil, T.: Vector-valued multifractal measures. Proc. Roy Soc. London Ser. A 452(1949), 1433–1457 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Forte, B., Mendivil, F., Vrscay, E.R.: IFS operators on integral transforms, in Fractals: Theory and Applications in Engineering. In: Dekking, M., Levy-Vehel, J., Lutton, E., Tricot, C. (eds.) . Springer Verlag, London (1999)

  12. Forte, B., Vrscay, E.R.: Theory of generalized fractal transforms, Fractal Image Encoding and Analysis, NATO ASI, Series, F, vol. 159. In: Fisher, Y. (ed.) Springer Verlag, New York (1998)

  13. Hermes, H.: Calculus of set-valued functions and control. J. Math. Mech. 18, 47–59 (1968/1969)

  14. Hildenbrand, W.: Core and Equilibria of a Large Economy, Princeton Univ. Press. Princeton, New York (1974)

    Google Scholar 

  15. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. J. Math. 30, 713–747 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Iacus, S.M., La Torre, D.: A comparative simulation study on the IFS distribution function estimator. Nonlinear Anal. Real World Appl. 6, 774–785 (2005)

    Article  MathSciNet  Google Scholar 

  17. Kandilakis, D.: On the extension of multimeasures and integration with respect to a multimeasure. Proc. AMS 116(1), 85–92 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kantorovich, L.V., Rubinstein, G.Sh.: On a functional space and certain extremal problems. Dokl. Akad, Nauk SSSR 115, 1058–1061 (1957). in Russian

  19. Kantorovich, L.V., Rubinstein, G.Sh.: On a space of completely additive functions. Vestnik. Leningrad. Univ. Math. 13, 52–59 (1958). (in Russian)

    MATH  Google Scholar 

  20. Koshevoy, G.: The Lorenz zonotope and multivariate majorizations. Soc. Choice Welf. 15(1), 1–14 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mendivil, F., Vrscay, E.R.: Fractal vector measures and vector calculus on planar fractal domains. Chaos Solitons Fractals 14, 1239–1254 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)

    Book  MATH  Google Scholar 

  23. Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester (1991)

    MATH  Google Scholar 

  24. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, New York (1998)

    Book  MATH  Google Scholar 

  25. Rockafellar, R.T.: Integral functional, normal integrands and measurable selectors, Lecture Notes in Math. N. 543. Springer Verlag Berlin (1976)

  26. Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  27. Vershik, A.M.: Kantorovich metric: Initial history and little-known applications. J. Math. Sci. 133(4), 1410–1417 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Villani, C.: Topics in optimal transportation. AMS (2003)

  29. Vind, K.: Edgeworth-allocations in an exchange economy with many traders. Int. Econ. Rev. 5(2), 165–177 (1964)

    Article  MATH  Google Scholar 

  30. WIllard, S.: General Topology. Addison-Wesley (1970)

Download references

Author information

Authors and Affiliations

  1. Department of Economics Management and Quantitative Methods, University of Milan, Milan, Italy

    D. La Torre

  2. Department of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi, UAE

    D. La Torre

  3. Department of Mathematics and Statistics, Acadia University, Nova Scotia, Wolfville, Canada

    F. Mendivil

Authors
  1. D. La Torre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Mendivil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Mendivil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

La Torre, D., Mendivil, F. The Monge–Kantorovich Metric on Multimeasures and Self–Similar Multimeasures. Set-Valued Var. Anal 23, 319–331 (2015). https://doi.org/10.1007/s11228-014-0310-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-014-0310-7

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation