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\(L_1\)-space for Sugeno Integral

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Book cover Modeling Decisions for Artificial Intelligence (MDAI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10571))

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Abstract

Let \(L_1(Su)\) be the \(L_1\) space with respect to the Sugeno integral for a fuzzy measure [7, 9]. \(L_1(Su)\) is a linear space with the natural quasi-metric. In general \(L_1(Su)\) is not necessarily a topological linear space. We shall characterize explicitely the maximal topological linear subspace of \(L_1(Su)\).

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References

  1. de Barra, G.: Introduction to Measure Theory. Van Nostrand Reinhold Company, New York (1974)

    Google Scholar 

  2. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  3. Denneberg, D.: Non-additive Measure and Integral. Theory and Decision Library B, vol. 27. Springer, Dordrecht (1994)

    Google Scholar 

  4. Frink, A.H.: Distance-functions and the metrization problem. Bull. Am. Math. Soc. 43, 133–142 (1937)

    Google Scholar 

  5. Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext, Springer (2001)

    Book  MATH  Google Scholar 

  6. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, Heidelberg (1965)

    Google Scholar 

  7. Honda, A., Okazaki, Y.: Quasi-metric on \(L_p\)-space for a fuzzy measure. In: 20th Workshop for Heart and Mind. Japan Society for Fuzzy Theory and Intelligent Informatics (SOFT) (2015)

    Google Scholar 

  8. Honda, A., Okazaki, Y.: \(L_p\)-space and its dual for a fuzzy measure. In: The 13th International Conference on Modelling Decisions for Artificial Intelligence (MDAI 2016), Andorra, 19–21 September 2016

    Google Scholar 

  9. Honda, A., Okazaki, Y.: Quasi-metrics on the function spaces \(L_p (0\,<\,p\,<\,\infty )\) for a fuzzy measure. In: RIMS Conference on the Structure of Function Spaces and its Related Fields, RIMS Kokyuroku (2017, to appear)

    Google Scholar 

  10. Honda, A., Okazaki, Y., Sato, H.: Approximations and the linearity of the Shepp space. Kyushu J. Math. 69(1), 173–194 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Murofushi, T., Sugeno, M.: A theory of fuzzy measures: representations, the choquet integral and null sets. JMAA 159, 534–549 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Okazaki, Y.: Topological linear subspace of \(L_0({\varOmega }, \mu )\) for the infinite measure \(\mu \). Nihonkai Math. J. 27, 145–154 (2016)

    Google Scholar 

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

    Google Scholar 

  14. Schroeder, V.: Quasi-metric and metric spaces, arXiv:math/0607304v1 [math:MG], 13 July 2006

  15. Sugeno, M., Murofushi, T.: Fuzzy Measure, Lectures on Fuzzy Measure, vol. 3. Nikkan Kogyo Shinbun LTD (1991). (in Japanese)

    Google Scholar 

  16. Triebel, H.: A new approach to function spaces on quasi-metric spaces. Revista Matematica Complutense 18(1), 7–48 (2005)

    Google Scholar 

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Acknowledgments

This work was supported by JSPS KAKENHI Grant number 26400155, 15K05003

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Authors and Affiliations

  1. Kyushu Institute of Technology, 680-2 Kawazu, Iizuka, Fukuoka, 820-8502, Japan

    Aoi Honda

  2. Fuzzy Logic Systems Institute, 680-41 Kawazu, Iizuka, Fukuoka, 820-0067, Japan

    Yoshiaki Okazaki

Authors
  1. Aoi Honda
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  2. Yoshiaki Okazaki
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Corresponding author

Correspondence to Aoi Honda .

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Editors and Affiliations

  1. University of Skövde, Skövde, Sweden

    Vicenç Torra

  2. Toho Gakuen, Kunitachi, Tokyo, Japan

    Yasuo Narukawa

  3. Kyushu Institute of Technology, Iizuka, Fukuoka, Japan

    Aoi Honda

  4. Kyushu Institute of Technology, Kitakyushu-shi, Fukuoka, Japan

    Sozo Inoue

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Honda, A., Okazaki, Y. (2017). \(L_1\)-space for Sugeno Integral. In: Torra, V., Narukawa, Y., Honda, A., Inoue, S. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2017. Lecture Notes in Computer Science(), vol 10571. Springer, Cham. https://doi.org/10.1007/978-3-319-67422-3_7

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  • DOI: https://doi.org/10.1007/978-3-319-67422-3_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67421-6

  • Online ISBN: 978-3-319-67422-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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