Soft Computing

, Volume 22, Issue 13, pp 4347–4351 | Cite as

The pseudo-convergence of measurable functions on set-valued fuzzy measure space



For sequences of measurable functions on a set-valued fuzzy measure space, the concepts of pseudo almost everywhere convergence, pseudo almost uniformly convergence, and pseudo-convergence in measure are introduced. Then, Egoroff’s theorem, Lebesgue’s theorem, and Riesz’s theorem are generalized from real-valued fuzzy measure spaces onto set-valued fuzzy measure spaces.


Fuzzy analysis Set-valued fuzzy measure Measurable function Convergence theorem 



Jianrong Wu has been supported by National Natural Science Foundation of China (No. 11371013). The authors acknowledge the reviewer’s comments and suggestions very much, which are valuable in improving the quality of our manuscript.

Compliance with ethical standards

Conflict of interest

Both authors declare that they have no conflict of interest.

Human and animals participants

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Artstein Z (1972) Set-valued measures. Trans Am Math Soc 165:103–121MathSciNetCrossRefMATHGoogle Scholar
  2. Denneberg D (1994) Non-additive measure and integral. Kluwer Academic Publishers, DordrechtCrossRefMATHGoogle Scholar
  3. Gao N, Li Y, Wang G (2008) Autocontinuity of set-valued fuzzy measures. J Sichuan Norm Univ 31:386–389 (in Chinese)MATHGoogle Scholar
  4. Gavrilut A (2009) Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions. Fuzzy Sets Syst 160:1308–1317MathSciNetCrossRefMATHGoogle Scholar
  5. Gavrilut A (2010a) Regularity and autocontinuity of set multifunctions. Fuzzy Sets Syst 161:681–693MathSciNetCrossRefMATHGoogle Scholar
  6. Gavrilut A (2010b) A Lusin type theorem for regular monotone uniformly autocontinuous set multifunctions. Fuzzy Sets Syst 161:2909–2918MathSciNetCrossRefMATHGoogle Scholar
  7. Gavrilut A (2013a) Abstract regular null-null-additive set multifunctions in Hausdorff topology. Ann Alexandru Ioan Cuza Univ Math 59:129–147Google Scholar
  8. Gavrilut A (2013b) Alexandroff theorem in Hausdorff topology for null-null-additive set multifunctions. Ann Alexandru Ioan Cuza Univ Math 59:237–251MathSciNetMATHGoogle Scholar
  9. Guo C, Zhang D (2004) On set-valued fuzzy measures. Inf Sci 160:13–25MathSciNetCrossRefMATHGoogle Scholar
  10. Hu S, Papageorgiou NS (1990) Handbook of multivalued analysis, vol 1. Kluwer Academic Publishers, DordrechtMATHGoogle Scholar
  11. Pap E (1995) Null-additive set functions. Kluwer Academic Publishers, DordrechtMATHGoogle Scholar
  12. Precupanu A, Gavrilut A (2011) A set-valued Egoroff type theorem. Fuzzy Sets Syst 175:87–95MathSciNetCrossRefMATHGoogle Scholar
  13. Precupanu A, Gavrilut A (2012a) Set-valued Lusin type theorem for null-null-additive set multifunctions. Fuzzy Sets Syst 204:106–116MathSciNetCrossRefMATHGoogle Scholar
  14. Precupanu A, Gavrilut A (2012b) Pseudo-convergences of sequences of measurable functions on monotone multimeasure spaces. Ann Alexandru Ioan Cuza Univ Math LVIII 58(1):67–84MathSciNetMATHGoogle Scholar
  15. Sugeno M (1974) Theory of fuzzy integrals and its applications. Dissertation, Tokyo Institute of TechnologyGoogle Scholar
  16. Wang ZY (1984) The autocontinuity of set function and the fuzzy integral. J Math Anal Appl 99:195–218MathSciNetCrossRefMATHGoogle Scholar
  17. Wang ZY (1985) Asymptotic structural characteristics of fuzzy measures and their applications. Fuzzy Sets Syst 16:277–290MathSciNetCrossRefMATHGoogle Scholar
  18. Wang ZY, Klir GJ (1992) Fuzzy measure theory. Plenum Press, New YorkCrossRefMATHGoogle Scholar
  19. Wang ZY, Klir GJ (2008) Generalized measure theory. Springer, New YorkMATHGoogle Scholar
  20. Wang ZY, Leung KS, Klir GJ (2006) Integration on finite sets. Int J Intell Syst 21:1073–1092CrossRefMATHGoogle Scholar
  21. Wang ZY, Yang R, Leung KS (2010) Nonlinear integrals and their applications in data mining. World Scientific, SingaporeCrossRefMATHGoogle Scholar
  22. Wu JR, Liu HY (2011) Autocontinuity of set-valued fuzzy measures and applications. Fuzzy Set Syst 175:57–64MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Physics ScienceSuzhou University of Science and TechnologySuzhouPeople’s Republic of China

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