Skip to main content
SpringerLink
Log in

A Study of Riesz Space-Valued Non-additive Measures

  • Chapter
Integrated Uncertainty Management and Applications

Part of the book series: Advances in Intelligent and Soft Computing ((AINSC,volume 68))

  • 937 Accesses

Abstract

This paper gives a short survey of our recent developments in Riesz space-valued non-additive measure theory and contains the following topics: the Egoroff theorem, the Lebesgue theorem, the Riesz theorem, the Lusin theorem, and the Alexandroff theorem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexandroff, A.D.: Additive set-functions in abstract spaces. Mat. Sbornik U.S. 9, 563–628 (1941)

    MATH  MathSciNet  Google Scholar 

  2. Denneberg, D.: Non-Additive Measure and Integral, 2nd edn. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  3. Dobrakov, I.: On submeasures I. Dissertationes Math (Rozprawy Mat.) 112, 1–35 (1974)

    MathSciNet  Google Scholar 

  4. Egoroff, D.-T.: Sur les suites des fonctions mesurables. C. R. Acad. Sci. Paris 152, 244–246 (1911)

    MATH  Google Scholar 

  5. Holbrook, J.A.R.: Seminorms and the Egoroff property in Riesz spaces. Trans. Amer. Math. Soc. 132, 67–77 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hrachovina, E.: A generalization of the Kolmogorov consistency theorem for vector measures. Acta Math. Univ. Comenian. 54-55, 141–145 (1988)

    MathSciNet  Google Scholar 

  7. Kawabe, J.: Uniformity for weak order convergence of Riesz space-valued measures. Bull. Austral. Math. Soc. 71, 265–274 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kawabe, J.: The Egoroff theorem for non-additive measures in Riesz spaces. Fuzzy Sets and Systems 157, 2762–2770 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kawabe, J.: The Egoroff property and the Egoroff theorem in Riesz space-valued non-additive measure theory. Fuzzy Sets and Systems 158, 50–57 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kawabe, J.: Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures. Fuzzy Sets and Systems 158, 895–903 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kawabe, J.: The Alexandroff theorem for Riesz space-valued non-additive measures. Fuzzy Sets and Systems 158, 2413–2421 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kawabe, J.: The countably subnormed Riesz space with applications to non-additive measure theory. In: Tsukada, M., Takahashi, W., Murofushi, M. (eds.) 2005 Symposium on Applied Functional Analysis, pp. 279–292. Yokohama Publishers, Yokohama (2007)

    Google Scholar 

  13. Kawabe, J.: The Choquet integral in Riesz space. Fuzzy Sets and Systems 159, 629–645 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kawabe, J.: Some properties on the regularity of Riesz space-valued non-additive measures. In: Kato, M., Maligranda, L. (eds.) Banach and function spaces II, pp. 337–348. Yokohama Publishers, Yokohama (2008)

    Google Scholar 

  15. Kawabe, J.: Continuity and compactness of the indirect product of two non-additive measures. Fuzzy Sets and Systems 160, 1327–1333 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kawabe, J.: Regularities of Riesz space-valued non-additive measures with applications to convergence theorems for Choquet integrals. Fuzzy Sets and Systems (2009); doi:10.1016/j.fss.2009.06.001

    Google Scholar 

  17. Li, J.: Order continuous of monotone set function and convergence of measurable functions sequence. Appl. Math. Comput. 135, 211–218 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Li, J.: On Egoroff’s theorems on fuzzy measure spaces. Fuzzy Sets and Systems 135, 367–375 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Li, J.: A further investigation for Egoroff’s theorem with respect to monotone set functions. Kybernetika 39, 753–760 (2003)

    MathSciNet  Google Scholar 

  20. Li, J., Yasuda, M.: Lusin’s theorem on fuzzy measure spaces. Fuzzy Sets and Systems 146, 121–133 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, J., Yasuda, M.: Egoroff’s theorem on monotone non-additive measure spaces. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 12, 61–68 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, J., Yasuda, M.: On Egoroff’s theorems on finite monotone non-additive measure space. Fuzzy Sets and Systems 153, 71–78 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, I. North-Holland, Amsterdam (1971)

    Google Scholar 

  24. Murofushi, T., Uchino, K., Asahina, S.: Conditions for Egoroff’s theorem in non-additive measure theory. Fuzzy Sets and Systems 146, 135–146 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht (1995)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Riečan, J.: On the Kolmogorov consistency theorem for Riesz space valued measures. Acta Math. Univ. Comenian. 48-49, 173–180 (1986)

    Google Scholar 

  28. Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer Academic Publishers, Bratislava (1997)

    MATH  Google Scholar 

  29. Schwartz, L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Bombay (1973)

    MATH  Google Scholar 

  30. Sugeno, M.: Theory of fuzzy integrals and its applications. Thesis, Tokyo Institute of Technology (1974)

    Google Scholar 

  31. Sun, Q.: Property (S) of fuzzy measure and Riesz’s theorem. Fuzzy Sets and Systems 62, 117–119 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Volauf, P.: Alexandrov and Kolmogorov consistency theorem for measures with values in partially ordered groups. Tatra Mt. Math. Publ. 3, 237–244 (1993)

    MATH  MathSciNet  Google Scholar 

  33. Wang, Z., Klir, G.J.: Fuzzy Measure Theory. Plenum Press, New York (1992)

    MATH  Google Scholar 

  34. Wright, J.D.M.: The measure extension problem for vector lattices. Ann. Inst. Fourier (Grenoble) 21, 65–85 (1971)

    MATH  MathSciNet  Google Scholar 

  35. Wu, C., Ha, M.: On the regularity of the fuzzy measure on metric fuzzy measure spaces. Fuzzy Sets and Systems 66, 373–379 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  36. Wu, J., Wu, C.: Fuzzy regular measures on topological spaces. Fuzzy Sets and Systems 119, 529–533 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan

    Jun Kawabe

Authors
  1. Jun Kawabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Knowledge Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, 923-1292, Ishikawa, Japan

    Van-Nam Huynh  & Yoshiteru Nakamori  & 

  2. Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, BS8 1TR, Bristol, United Kingdom

    Jonathan Lawry

  3. Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, 560-8531, Osaka, Japan

    Masahiro Inuiguchi

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kawabe, J. (2010). A Study of Riesz Space-Valued Non-additive Measures. In: Huynh, VN., Nakamori, Y., Lawry, J., Inuiguchi, M. (eds) Integrated Uncertainty Management and Applications. Advances in Intelligent and Soft Computing, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11960-6_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-11960-6_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11959-0

  • Online ISBN: 978-3-642-11960-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation