{"title":"单调度量的拉顿-尼科迪姆定理","authors":"Yao Ouyang , Jun Li","doi":"10.1016/j.fss.2024.108995","DOIUrl":null,"url":null,"abstract":"<div><p>A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair <span><math><mo>(</mo><mi>μ</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function <em>f</em>. If <em>ν</em> is null-continuous and weakly null-additive, then <em>f</em> is uniquely determined almost everywhere by <em>ν</em> and thus is called the Radon-Nikodym derivative of <em>μ</em> w.r.t. <em>ν</em>. For <em>σ</em>-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.</p></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Radon-Nikodym theorem for monotone measures\",\"authors\":\"Yao Ouyang , Jun Li\",\"doi\":\"10.1016/j.fss.2024.108995\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair <span><math><mo>(</mo><mi>μ</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function <em>f</em>. If <em>ν</em> is null-continuous and weakly null-additive, then <em>f</em> is uniquely determined almost everywhere by <em>ν</em> and thus is called the Radon-Nikodym derivative of <em>μ</em> w.r.t. <em>ν</em>. For <em>σ</em>-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.</p></div>\",\"PeriodicalId\":55130,\"journal\":{\"name\":\"Fuzzy Sets and Systems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2024-05-03\",\"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/S0165011424001416\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424001416","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}
A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function f. If ν is null-continuous and weakly null-additive, then f is uniquely determined almost everywhere by ν and thus is called the Radon-Nikodym derivative of μ w.r.t. ν. For σ-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.
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.