{"title":"The continuous Choquet integral representation theorems in an abstract setting","authors":"Jun Kawabe","doi":"10.1016/j.fss.2024.109115","DOIUrl":null,"url":null,"abstract":"<div><p>The purpose of the paper is to formulate Choquet integral representation theorems for a monotone functional on a collection of functions in such a way that the representing measures are simultaneously inner and outer continuous on appropriate collections of sets such as open, closed, compact, and measurable. This type of theorem is referred to as the <em>continuous</em> Choquet integral representation theorem and will be discussed in a setting general enough for practical use. The benefits of our results are as follows:</p><ul><li><span>(i)</span><span><p>the representing measures are simultaneously inner and outer continuous,</p></span></li><li><span>(ii)</span><span><p>the collections of sets for which the representing measures are inner and outer continuous are larger than those in previous studies,</p></span></li><li><span>(iii)</span><span><p>the regularity of the representing measures is also considered, and</p></span></li><li><span>(iv)</span><span><p>it is possible to handle not only <em>σ</em>-continuous but also <em>τ</em>-continuous functionals.</p></span></li></ul></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424002616","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
The purpose of the paper is to formulate Choquet integral representation theorems for a monotone functional on a collection of functions in such a way that the representing measures are simultaneously inner and outer continuous on appropriate collections of sets such as open, closed, compact, and measurable. This type of theorem is referred to as the continuous Choquet integral representation theorem and will be discussed in a setting general enough for practical use. The benefits of our results are as follows:
(i)
the representing measures are simultaneously inner and outer continuous,
(ii)
the collections of sets for which the representing measures are inner and outer continuous are larger than those in previous studies,
(iii)
the regularity of the representing measures is also considered, and
(iv)
it is possible to handle not only σ-continuous but also τ-continuous functionals.
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.
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