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Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

Author

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  • Leifan Yan

    (Key Laboratory of Media Audio and Video of the Ministry of Education, Communication University of China, Beijing 100024, China
    School of Data Science and Media Intelligence, Communication University of China, Beijing 100024, China)

  • Tong Kang

    (School of Data Science and Media Intelligence, Communication University of China, Beijing 100024, China
    School of Sciences, Communication University of China, Beijing 100024, China)

  • Huai Zhang

    (Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China)

Abstract

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented.

Suggested Citation

  • Leifan Yan & Tong Kang & Huai Zhang, 2023. "Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures," Mathematics, MDPI, vol. 11(13), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:3013-:d:1188534
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    References listed on IDEAS

    as
    1. Ehud Lehrer, 2009. "A new integral for capacities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(1), pages 157-176, April.
    2. Yaarit Even & Ehud Lehrer, 2014. "Decomposition-integral: unifying Choquet and the concave integrals," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(1), pages 33-58, May.
    3. Anca Croitoru & Alina Gavriluţ & Alina Iosif & Anna Rita Sambucini, 2022. "Convergence Theorems in Interval-Valued Riemann–Lebesgue Integrability," Mathematics, MDPI, vol. 10(3), pages 1-15, January.
    4. Ehud Lehrer & Roee Teper, 2020. "Set-valued capacities: multi-agenda decision making," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(1), pages 233-248, February.
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