Skip to main content
Springer Nature Link
Log in

On Weak Null-Additivity of Monotone Measures

  • Conference paper
Advances in Computational Intelligence (IPMU 2012)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 300))

  • 1302 Accesses

Abstract

In this note, the relations between weak null-additivity and pseudometric generating property of monotone measures are discussed. We show that on finite continuous monotone measure spaces \((X, {\cal F}, \mu)\), if measurable space \((X, {\cal F})\) is S-compact (especially, if X is countable), then the weak null-additivity is equivalent to pseudometric generating property. We put a question: abandoning the S-compactness condition, does the equivalence remain valid?

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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.

Similar content being viewed by others

References

  1. Asahina, S., Uchino, K., Murofushi, T.: Relationship among continuity conditions and null-additivity conditions in non-additive measure theory. Fuzzy Sets and Systems 157(2), 691–698 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dobrakov, I., Farkova, J.: On submeasures II. Math. Slovaca 30, 65–81 (1980)

    MathSciNet  MATH  Google Scholar 

  3. Jiang, Q., Wang, S., Ziou, D.: A further investigation for fuzzy measures on metric spaces. Fuzzy Sets and Systems 105(1), 293–297 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jiang, Q., Wang, S., Ziou, D., Wang, Z., Klir, G.J.: Pseudometric generated preporty and autocontinuity of fuzzy measure. Fuzzy Sets and Systems 112(2), 207–216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kawabe, J.: Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures. Fuzzy Sets and Systems 158(8), 895–903 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kawabe, J.: The Alexandroff theorem for Riesz space-valued non-additive measures. Fuzzy Sets and Systems 158(21), 2413–2421 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kawabe, J.: Continuity and compactness of the indirect product of two non-additive measures. Fuzzy Sets and Systems 160(9), 1327–1333 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kawabe, J.: Regularities of Riesz space-valued non-additive measures with applications to convergence theorems for Choquet integrals. Fuzzy Sets and Systems 161(5), 642–650 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, J., Yasuda, M., Jiang, Q., Suzuki, H., Wang, Z., Klir, G.J.: Convergence of sequence of measurable functions on fuzzy measure space. Fuzzy Sets and Systems 87(3), 385–387 (1997)

    Article  MathSciNet  Google Scholar 

  10. Li, J., Zhang, Q.: Asymptotic Structural Characteristics of Monotone Measure and Convergence in Monotone measure. The J. of Fuzzy Math. 9(2), 447–459 (2001)

    MATH  Google Scholar 

  11. Li, J.: Order continuous of monotone set function and convergence of measurable functions sequence. Applied Mathematics and Computation 135(2-3), 211–218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, J., Yasuda, M.: Lusin’s theorem on fuzzy measure spaces. Fuzzy Sets and Systems 146(1), 121–133 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, J., Yasuda, M.: On Egoroff’s theorem on finite monotone non-additive measure space. Fuzzy Sets and Systems 153(1), 71–78 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, J., Mesiar, R.: Lusin’s theorem on monotone measure spaces. Fuzzy Sets and Systems 175(1), 75–86 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MathSciNet  MATH  Google Scholar 

  16. Murofushi, T.: Extensions of (weakly) null-additive, monotone set functions from rings of subsets to generated algebras. Fuzzy Sets and Systems 158(21), 2422–2428 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Narukawa, Y., Murofushi, T.: Choquet integral with respect to a regular non-additive measure. In: Proceedings of FUZZ-IEEE 2004 International Conference, vol. 1, pp. 517–521 (2004)

    Google Scholar 

  18. Pap, E.: Null-additive Set Functions. Kluwer Academic Press, Dordrecht (1995)

    MATH  Google Scholar 

  19. Precupanu, A., Gavrilut, A., Croitoru, A.: A set-valued Egoroff type theorem. Fuzzy Sets and Systems 175(1), 87–95 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Uchino, K., Murofushi, T.: Relations between mathematical properties of fuzzy measures. In: 10th IFSA World Congress, Istanbul, Turkey, pp. 27–30 (2003)

    Google Scholar 

  21. Wang, Z.: Asymptotic structural characteristics of fuzzy measure and their applications. Fuzzy Sets and Systems 16, 277–290 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer (2009)

    Google Scholar 

  23. Watanabe, T., Kawasaki, T., Tanaka, T.: On a sufficient condition of Lusin’s theorem for non-additive measures that take values in an ordered topological vector space. Fuzzy Sets and Systems 194, 66–75 (2012)

    Article  Google Scholar 

  24. Wu, C., Sun, B.: Pseudo-atoms of fuzzy and non-fuzzy measure. Fuzzy Sets and Systems 158(11), 1258–1272 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, Q., Xu, Y., Du, W.: Lebesgue decomposition theorem for σ-finite signed fuzzy measures. Fuzzy Sets and Systems 101(3), 445–451 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Communication University of China, Beijing, 100024, China

    Jun Li

  2. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, SK-81368, Bratislava, Slovakia

    Radko Mesiar

  3. College of Mathematics and Informational Technology, Xinjiang Educational Institute, Urumqi, Xinjiang, 830043, China

    Hemin Wu

Authors
  1. Jun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radko Mesiar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hemin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Economics, University of Catania, Corso Italia, 55, 95129, Catania, Italy

    Salvatore Greco  & Benedetto Matarazzo  & 

  2. CNRS UMR 7606, DAPA, LIP6 8, Université Pierre et Marie Curie - Paris6, rue du Capitaine Scott, F-75015, Paris, France

    Bernadette Bouchon-Meunier

  3. Dip. Matematica e Informatica, Università di Perugia, 06123, Perugia, Italy

    Giulianella Coletti

  4. Department of Computer and Management Science, University of Trento, Via Inama 5, 38122, Trento, Italy

    Mario Fedrizzi

  5. Machine Intelligence Institute — IONA College,, 10801, New Rochelle, NY, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Li, J., Mesiar, R., Wu, H. (2012). On Weak Null-Additivity of Monotone Measures. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31724-8_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31724-8_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31723-1

  • Online ISBN: 978-3-642-31724-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics