集值函数关于模糊测度Choquet积分的表示和积分原函数性质

doi:10.6040/j.issn.1671-9352.0.2016.450

山东大学学报（理学版） ›› 2017, Vol. 52 ›› Issue (8): 1-9.doi: 10.6040/j.issn.1671-9352.0.2016.450

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集值函数关于模糊测度Choquet积分的表示和积分原函数性质

巩增泰,寇旭阳

西北师范大学数学与统计学院, 甘肃兰州 730070

收稿日期:2016-09-23 出版日期:2017-08-20 发布日期:2017-08-03
作者简介:巩增泰(1966— )男, 博士, 教授, 研究方向为实分析、模糊分析学、粗糙集理论. E-mail:gongzt@nwnu.edu.cn
基金资助:
国家自然科学基金资助项目(11461062,61262022)

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

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

Received:2016-09-23 Online:2017-08-20 Published:2017-08-03

摘要/Abstract

摘要： 研究了集值函数关于模糊测度Choquet积分的分析性质: 讨论了集值函数Choquet积分的计算方法, 给出了集值函数Choquet积分的表示定理和Radon-Nikodym性质, 并且对集值函数Choquet积分的原函数进行了刻划。最后, 对集值函数关于模糊测度Choquet积分定义进行了改进, 提出了集值函数 “上方函数” 和 “下方函数” 概念, 实现了对集值函数关于模糊测度的Choquet积分的控制。

关键词: 模糊测度, Choquet积分, 集值函数

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

中图分类号:

O175.8

巩增泰,寇旭阳. 集值函数关于模糊测度Choquet积分的表示和积分原函数性质[J]. 山东大学学报（理学版）, 2017, 52(8): 1-9.

GONG Zeng-tai, KOU Xu-yang. Representation of Choquet integral of the set-valued functions with respect to fuzzy measures and the characteristic of its primitive[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 1-9.

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集值函数关于模糊测度Choquet积分的表示和积分原函数性质

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

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