Some notes on the coincidence of the Choquet integral and the pan-integral

IF 3.2 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Fuzzy Sets and Systems Pub Date : 2024-11-05 DOI:10.1016/j.fss.2024.109178

Tong Kang , Radko Mesiar , Yao Ouyang , Jun Li

引用次数: 0

Abstract

In this note, we provide an example to show the weak (M)-property is really weaker than the (M)-property for some finite monotone measure defined on infinite space, and hence answer an open problem which was proposed in the paper (Li et al. (2023) [9]). We prove that if a monotone measure μ is autocontinuous, then the weak (M)-property and the (M)-property of μ are equivalent. We propose the concept of (C-P)-property of monotone measures and show a set of sufficient and necessary conditions that the Choquet integral coincides with the pan-integral. We further study the relationships between the Choquet integral and the pan-integral in the setting of the ordered pairs of monotone measures, and obtain some interesting properties.

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关于乔奎特积分和泛积分重合的一些说明

在本注中，我们举例说明对于定义在无限空间上的某个有限单调度量，弱（M）属性确实比（M）属性弱，从而回答了论文（李等（2023）[9]）中提出的一个开放问题。我们证明，如果单调度量 μ 是自连续的，那么弱 (M)- 属性和 μ 的 (M)- 属性是等价的。我们提出了单调度量的(C-P)-属性概念，并证明了一组乔奎特积分与泛积分重合的充分必要条件。我们进一步研究了在单调度量的有序对的背景下，Choquet积分和泛积分之间的关系，并得到了一些有趣的性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Fuzzy Sets and Systems 数学-计算机：理论方法

CiteScore

6.50

自引率

17.90%

发文量

321

审稿时长

6.1 months

期刊介绍： Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies. In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.