JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (8): 1-9.doi: 10.6040/j.issn.1671-9352.0.2016.450

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Representation of Choquet integral of the set-valued functions with respect to fuzzy measures and the characteristic of its primitive

GONG Zeng-tai, KOU Xu-yang   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2016-09-23 Online:2017-08-20 Published:2017-08-03

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Abstract: The analytic properties of the Choquet integral of set-valued functions with respect to fuzzy measures are discussed, such as the characteristics of the primitive, representation of integral, differentiability of the primitive, and so on. Firstly, based on the previous results, the calculation of Choquet integral of set-valued function is investigated, and a representation theorem of Choquet integral for set-valued function is obtained as a Radon-Nikodym property in some sense. In addition, the characteristics of the primitive of the Choquet integral for set valued functions are given. Finally, the definition of the Choquet integral of set-valued functions with respect to fuzzy measures is improved, and the concepts of the above functions and below function of the set-valued functions are proposed, which achieved the domination of the Choquet integral of set-valued functions with respect to fuzzy measures.

Key words: set-valued functions, fuzzy measure, Choquet integral

CLC Number: 

  • O175.8
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