Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

Leifan Yan, Tong Kang () and Huai Zhang
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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
Tong Kang: School of Data Science and Media Intelligence, Communication University of China, Beijing 100024, China
Huai Zhang: Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China

Mathematics, 2023, vol. 11, issue 13, 1-14

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.

Keywords: set-valued function; fuzzy measure; decomposition integral; choquet integral; pan-integral; concave integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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