Skip to main content
SpringerLink
Log in

An Equivalent Definition of Pan-Integral

  • Conference paper
  • First Online:
Modeling Decisions for Artificial Intelligence (MDAI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9880))

Abstract

In this note, we introduce the concepts of support disjointness super-\(\oplus \)-additivity and positively super-\(\otimes \)-homogeneity of a functional (with respect to pan-addition \(\oplus \) and pan-multiplication \(\otimes \), respectively). By means of these two properties of functionals, we discuss the characteristics of pan-integrals and present an equivalent definition of the pan-integral. As special cases, we obtain the equivalent definitions of the Shilkret integral, the \(+,\cdot \)-based pan-integral, and the Sugeno integral.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

References

  1. Benvenuti, P., Mesiar, R., Vivona, D.: Monotone set functions-based integrals. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1329–1379. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  2. Benvenuti, P., Mesiar, R.: Pseudo-arithmetical operations as a basis for the general measure and integration theory. Inf. Sci. 160, 1–11 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals: Theory and Applications. Studies in Fuzziness Soft Computing, vol. 40. Physica, Heidelberg (2000)

    MATH  Google Scholar 

  5. Ichihashi, H., Tanaka, M., Asai, K.: Fuzzy integrals based on pseudo-additions and multiplications. J. Math. Anal. Appl. 130, 354–364 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lehrer, E.: A new integral for capacities. Econ. Theor. 39, 157–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lehrer, E., Teper, R.: The concave integral over large spaces. Fuzzy Sets Syst. 159, 2130–2144 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mesiar, R.: Choquet-like integrals. J. Math. Anal. Appl. 194, 477–488 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mesiar, R., Rybárik, J.: Pan-operaions structure. Fuzzy Sets Syst. 74, 365–369 (1995)

    Article  MATH  Google Scholar 

  10. Pap, E.: Null-Additive Set Functions. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

  11. Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. dissertation, Takyo Institute of Technology (1974)

    Google Scholar 

  13. Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Teper, R.: On the continuity of the concave integral. Fuzzy Sets Syst. 160, 1318–1326 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tong, X., Chen, M., Li, H.X.: Pan-operations structure with non-idempotent pan-addition. Fuzzy Sets Syst. 145, 463–470 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  17. Yang, Q.: The pan-integral on fuzzy measure space. Fuzzy Math. 3, 107–114 (1985). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  18. Zhang, Q., Mesiar, R., Li, J., Struk, P.: Generalized Lebesgue integral. Int. J. Approx. Reason. 52, 427–443 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the National Natural Science Foundation of China (Grant No. 11371332 and No. 11571106) and the NSF of Zhejiang Province (No. LY15A010013).

Author information

Authors and Affiliations

  1. Faculty of Science, Huzhou University, Huzhou, 313000, Zhejiang, China

    Yao Ouyang

  2. School of Sciences, Communication University of China, Beijing, 100024, China

    Jun Li

Authors
  1. Yao Ouyang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yao Ouyang .

Editor information

Editors and Affiliations

  1. University of Kövde , Skövde, Sweden

    Vicenç Torra

  2. Toho Gakuen , Kunitachi, Tokyo, Japan

    Yasuo Narukawa

  3. Enginyeria de la Informacio I de les Com, Univ Autonoma de Barcelona, Bellaterra, Spain

    Guillermo Navarro-Arribas

  4. Universitat d'Andorra , Sant Julià de Lòria, Andorra

    Cristina Yañez

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Ouyang, Y., Li, J. (2016). An Equivalent Definition of Pan-Integral. In: Torra, V., Narukawa, Y., Navarro-Arribas, G., Yañez, C. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2016. Lecture Notes in Computer Science(), vol 9880. Springer, Cham. https://doi.org/10.1007/978-3-319-45656-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45656-0_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45655-3

  • Online ISBN: 978-3-319-45656-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation