
Representation of Choquet integral of the setvalued functions with respect to fuzzy measures and the characteristic of its primitive
 GONG Zengtai, KOU Xuyang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):
19.
doi:10.6040/j.issn.16719352.0.2016.450

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The analytic properties of the Choquet integral of setvalued 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 setvalued function is investigated, and a representation theorem of Choquet integral for setvalued function is obtained as a RadonNikodym 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 setvalued functions with respect to fuzzy measures is improved, and the concepts of the above functions and below function of the setvalued functions are proposed, which achieved the domination of the Choquet integral of setvalued functions with respect to fuzzy measures.