Advances in Mathematical Economics Volume 19 pp 1-32

Part of the Advances in Mathematical Economics book series (MATHECON, volume 19) | Cite as

On the Integration of Fuzzy Level Sets

  • Charles Castaing
  • Christiane Godet-Thobie
  • Thi Duyen Hoang
  • P. Raynaud de Fitte

Abstract

We study the integration of fuzzy level sets associated with a fuzzy random variable when the underlying space is a separable Banach space or a weak star dual of a separable Banach space. In particular, the expectation and the conditional expectation of fuzzy level sets in this setting are presented. We prove the SLLN for pairwise independent identically distributed fuzzy convex compact valued level sets through the SLLN for pairwise independent identically distributed convex compact valued random set in separable Banach space. Some convergence results for a class of integrand martingale are also presented.

Keywords

Conditional expectation Fuzzy convex Fuzzy martingale Integrand martingale Level set Upper semicontinuous 

References

  1. 1.
    Artstein Z, Hansen JC (1985) Convexification in limit laws of random sets in Banach spaces. Ann Probab 13(1):307–309MathSciNetCrossRefGoogle Scholar
  2. 2.
    Castaing C (1970) Quelques résultats de compacité liés à l’intégration. C R Acad Sci Paris Sér A-B 270:A1732–A1735. Actes du Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1971), pp 73–81. Bull. Soc. Math. France, Mém. No. 31–32. Soc. Math. France, Paris (1972)Google Scholar
  3. 3.
    Castaing C (2011) Some various convergence results for normal integrands. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 15. Springer, Tokyo, pp 1–26CrossRefGoogle Scholar
  4. 4.
    Castaing C, Ezzaki F, Lavie M, Saadoune M (2011) Weak star convergence of martingales in a dual space. In: Hudzik H, Lewicki G, Musielak J, Nowak M, Skrzypczak L (eds) Function spaces IX. Banach center publications, vol 92. Polish Academy of Sciences, Institute of Mathematics, Warsaw, pp 45–73Google Scholar
  5. 5.
    Castaing C, Hess C, Saadoune M (2008) Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions. In: Kusuoka S, Yamazaki A (eds) Advances in mathematical economics, vol 11. Springer, Tokyo, pp 11–44Google Scholar
  6. 6.
    Castaing C, Raynaud de Fitte P (2013) Law of large numbers and ergodic theorem for convex weak star compact valued Gelfand-integrable mappings. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 17. Springer, Tokyo, pp 1–37Google Scholar
  7. 7.
    Castaing C, Raynaud de Fitte P, Valadier M (2004) Young measures on topological spaces. With applications in control theory and probability theory. Mathematics and its applications, vol 571. Kluwer Academic, DordrechtGoogle Scholar
  8. 8.
    Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Lecture notes in mathematics, vol 580. Springer, Berlin/New YorkGoogle Scholar
  9. 9.
    de Blasi FS, Tomassini L (2011) On the strong law of large numbers in spaces of compact sets. J Convex Anal 18(1):285–300MathSciNetGoogle Scholar
  10. 10.
    Feron R (1976) Ensembles aléatoires flous. C R Acad Sci Paris Sér A-B 282(16):Aiii, A903–A906Google Scholar
  11. 11.
    Fitzpatrick S, Lewis AS (2006) Weak-star convergence of convex sets. J Convex Anal 13(3–4):711–719MathSciNetGoogle Scholar
  12. 12.
    Hiai F (1984) Strong laws of large numbers for multivalued random variables. In: Multifunctions and integrands (Catania, 1983). Lecture notes in mathematics, vol 1091. Springer, Berlin, pp 160–172Google Scholar
  13. 13.
    Hiai F, Umegaki H (1977) Integrals, conditional expectations, and martingales of multivalued functions. J Multivar Anal 7(1):149–182MathSciNetCrossRefGoogle Scholar
  14. 14.
    Höhle U, Šostak AP (1999) Axiomatic foundations of fixed-basis fuzzy topology. In: Höhle U, Rodabaugh SE (eds) Mathematics of fuzzy sets. Handbooks of fuzzy sets series, vol 3. Kluwer Academic, Boston, pp 123–272CrossRefGoogle Scholar
  15. 15.
    Inoue H (1991) A strong law of large numbers for fuzzy random sets. Fuzzy Sets Syst 41(3):285–291CrossRefGoogle Scholar
  16. 16.
    Jalby V (1992) Semi-continuité, convergence et approximation des applications vectorielles. loi des grands nombres. Technical report, Université Montpellier II, Laboratoire Analyse Convexe, 34095 Montpellier Cedex 05, FranceGoogle Scholar
  17. 17.
    Joo SY, Kim YK, Kwon JS (2006) Strong convergence for weighted sums of fuzzy random sets. Inf Sci 176(8):1086–1099MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kruse R (1982) The strong law of large numbers for fuzzy random variables. Inf Sci 28(3):233–241MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kwakernaak H (1978) Fuzzy random variables. I. Definitions and theorems. Inform Sci 15(1):1–29MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kwakernaak H (1979) Fuzzy random variables. II. Algorithms and examples for the discrete case. Inform Sci 17(3):253–278MathSciNetGoogle Scholar
  21. 21.
    Li S, Ogura Y (1996) Fuzzy random variables, conditional expectations and fuzzy valued martingales. J Fuzzy Math 4(4):905–927MathSciNetGoogle Scholar
  22. 22.
    Li S, Ogura Y (2006) Strong laws of large numbers for independent fuzzy set-valued random variables. Fuzzy Sets Syst 157(19):2569–2578MathSciNetCrossRefGoogle Scholar
  23. 23.
    Molchanov IS (1999) On strong laws of large numbers for random upper semicontinuous functions. J Math Anal Appl 235(1):349–355MathSciNetCrossRefGoogle Scholar
  24. 24.
    Neveu J (1972) Martingales à temps discret. Masson et Cie, éditeurs, ParisGoogle Scholar
  25. 25.
    Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114(2):409–422MathSciNetCrossRefGoogle Scholar
  26. 26.
    Puri ML, Ralescu DA (1991) Convergence theorem for fuzzy martingales. J Math Anal Appl 160(1):107–122MathSciNetCrossRefGoogle Scholar
  27. 27.
    Stojaković M (1994) Fuzzy random variables, expectation, and martingales. J Math Anal Appl 184(3):594–606MathSciNetCrossRefGoogle Scholar
  28. 28.
    Valadier M (1975) Convex integrands on Souslin locally convex spaces. Pac J Math 59(1):267–276MathSciNetCrossRefGoogle Scholar
  29. 29.
    Valadier M (1980) On conditional expectation of random sets. Ann Mat Pura Appl (4) 126:81–91Google Scholar
  30. 30.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Charles Castaing
    • 1
  • Christiane Godet-Thobie
    • 2
  • Thi Duyen Hoang
    • 3
  • P. Raynaud de Fitte
    • 4
  1. 1.Département de Mathématiques, Case courrier 051Université Montpellier IIMontpellier Cedex 5France
  2. 2.Laboratoire de Mathématiques de Bretagne AtlantiqueUniversité de BrestBrest Cedex3France
  3. 3.Quang Binh UniversityQuang BinhViet Nam
  4. 4.Laboratoire Raphaël Salem, UFR SciencesUniversité de Rouen, UMR CNRS 6085Saint Etienne du RouvrayFrance

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