• Inés CousoEmail author
  • Didier Dubois
  • Luciano Sánchez
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


Random sets originate in works published in the mid-sixties by well-known economists, Aumann [1] and Debreu [3] on the integration of set-valued functions. They have been given a full-fledged mathematical development by Kendall [10] and Matheron [16]. Random sets seem to have been originally used to handle uncertainty in spatial information, namely to tackle uncertainty in the definitions of geographical areas, in mathematical morphology, and in connection to geostatistics (to which Matheron is a major pioneering contributor, as seen by his work on kriging). Under this view, a precise realisation of a random set process is a precisely located set or region in an area of interest. This approach, especially applied to continuous spaces, raises subtle mathematical issues concerning the correct topology for handling set-valued realisations, that perhaps hide the intrinsic simplicity and intuitions behind random sets (e.g., casting them in a finite setting). The reason is that continuous random sets, like in geostatistics, were perhaps more easily found in applications than finite ones at that time. In any case this peculiarity has confined random sets to very specialised areas of mathematics.


Belief Function Evidence Theory Fuzzy Random Variable Possibility Theory Fuzzy Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    J. Aumann, Integral of set valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Colubi, R. Coppi, P. D’Urso, M.A. Gil. Statistics with fuzzy random variables. METRON-Int. J. Stat. LXV, 277–303 (2007)Google Scholar
  3. 3.
    G. Debreu, Integration of correspondences, in Proceedings of the Fifth Berkeley Symposium of Mathematical Statistics and Probability (Berkeley, USA, 1965), pp. 351–372Google Scholar
  4. 4.
    A.P. Dempster, Upper and lower probabilities induced by a multi-valued mapping. Ann. Math. Stat. 38, 325–339 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    T. Denœux, Modeling vague beliefs using fuzzy-valued belief structures. Fuzzy Sets Syst. 116(2), 167–199 (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980)zbMATHGoogle Scholar
  7. 7.
    D. Dubois, H. Prade, Evidence measures based on fuzzy information. Automatica 21(5), 547–562 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D. Dubois, H. Prade, Possibility Theory (Plenum Press, New York, 1988)CrossRefzbMATHGoogle Scholar
  9. 9.
    R. Féron, Ensembles aléatoires flous. C.R. Acad. Sci. Paris Ser. A 282, 903–906 (1976)Google Scholar
  10. 10.
    D.G. Kendall, Foundations of a theory of random sets, in Stochastic Geometry, ed. by E.F. Harding, D.G. Kendall (Wiley, New York, 1974), pp. 322–376Google Scholar
  11. 11.
    R. Körner, On the variance of fuzzy random variables. Fuzzy Sets Syst. 92, 83–93 (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    R. Kruse, On the variance of random sets. J. Math. Anal. Appl. 122, 469–473 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    R. Kruse, K.D. Meyer, Statistics with Vague Data (D. Reidel Publishing Company, Dordrecht, 1987)CrossRefzbMATHGoogle Scholar
  14. 14.
    H. Kwakernaak, Fuzzy random variables: definition and theorems. Inf. Sci. 15, 1–29 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    H. Kwakernaak, Fuzzy random variables: algorithms and examples in the discrete case. Inform. Sci. 17, 253–278 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    G. Matheron, Random Sets and Integral Geometry (Wiley, New York, 1975)zbMATHGoogle Scholar
  17. 17.
    H.T. Nguyen, On random sets and belief functions. J. Math. Anal. Appl. 63, 531–542 (1978)CrossRefGoogle Scholar
  18. 18.
    M. Puri, D. Ralescu, Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, 1976)zbMATHGoogle Scholar
  20. 20.
    G.L.S. Shackle, Decision, Order and Time in Human Affairs, 2nd edn. (Cambridge University Press, UK, 1961)Google Scholar
  21. 21.
    C.A.B. Smith, Consistency in statistical inference and decision. J. R. Stat. Soc. B 23, 1–37 (1961)zbMATHGoogle Scholar
  22. 22.
    P. Walley, Measures of uncertainty in expert systems. Artif. Intell. 83(1), 1–58 (1996)CrossRefMathSciNetGoogle Scholar
  23. 23.
    R.R. Yager, A foundation for a theory of possibility. Cybern. Syst. 10(1–3), 177–204 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    R.R. Yager, An introduction to applications of possibility theory. Hum. Syst. Manag. 3, 246–269 (1983)Google Scholar
  25. 25.
    R.R. Yager, An introduction to applications of possibility theory. Hum. Syst. Manag. 3, 246–269 (1983)Google Scholar
  26. 26.
    L.A. Zadeh, Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    L.A. Zadeh, Fuzzy sets and information granularity, in Advances in Fuzzy Set Theory and Applications, ed. by M.M. Gupta, R.K. Ragade, R.R. Yager. (North-Holland, Amsterdam, 1979), pp. 3–18Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Departamento de Estadística e I. O. y D. M.Universidad de OviedoOviedoSpain
  2. 2.IRIT, CNRSUniversité Paul SabatierToulouse Cedex 09France
  3. 3.Departamento de InformáticaUniversidad de OviedoOviedoSpain

Personalised recommendations