{"title":"A Note on Generalized Vitali Sets with Respect to Some Arbitrary Deformed Sums","authors":"Brian Villegas-Villalpando, Jorge E. Macías-Díaz","doi":"10.1016/S0034-4877(22)00082-9","DOIUrl":null,"url":null,"abstract":"<div><p><span><span>In this manuscript, we present a generalized deformed sum inspired by the nonadditive property of entropies such as those investigated by Tsallis and Shannon in the context of information theory. From this deformed sum, we define a generalization of the </span>Vitali set and prove its nonmeasurability. Moreover, the standard sum is recovered as the deforming parameter tends to zero, and Vitali's theorem is retrieved. In particular, the present work is a generalization of the results derived in </span><span>[2]</span><span> for arbitrary functions satisfying general conditions, and not only nonextensive statistics.</span></p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"90 3","pages":"Pages 377-383"},"PeriodicalIF":1.0000,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports on Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0034487722000829","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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Abstract
In this manuscript, we present a generalized deformed sum inspired by the nonadditive property of entropies such as those investigated by Tsallis and Shannon in the context of information theory. From this deformed sum, we define a generalization of the Vitali set and prove its nonmeasurability. Moreover, the standard sum is recovered as the deforming parameter tends to zero, and Vitali's theorem is retrieved. In particular, the present work is a generalization of the results derived in [2] for arbitrary functions satisfying general conditions, and not only nonextensive statistics.
期刊介绍：
Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.
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