{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T18:59:44Z","timestamp":1781204384485,"version":"3.54.1"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T00:00:00Z","timestamp":1723507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801307"],"award-info":[{"award-number":["11801307"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2017MA012"],"award-info":[{"award-number":["ZR2017MA012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2021MA009"],"award-info":[{"award-number":["ZR2021MA009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["11801307"],"award-info":[{"award-number":["11801307"]}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["ZR2017MA012"],"award-info":[{"award-number":["ZR2017MA012"]}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["ZR2021MA009"],"award-info":[{"award-number":["ZR2021MA009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to study its properties. Lusin\u2019s theorem is one of the most classical theorems in measure theory that reveals the close relationship between measurable and continuous functions, and has important significance. In this paper, we give three pairs of continuity conditions for uncertain measures, and present that every pair reveals duality, which is a kind of symmetry between objects. Furthermore, it is demonstrated that these continuity conditions are equivalent. And, we also prove that these three pairs of continuity conditions and the condition: if {\u039bn} is a sequence of open sets and \u039bn\u2198\u2205, then limn\u2192\u221eM{\u039bn}=0 are equivalent in compact metric spaces. It is shown that Lusin\u2019s theorem for uncertain variables holds if and only if the uncertain measure satisfies any of the above continuity conditions in a compact metric space. And, Lusin\u2019s theorem can be applied to uncertain variables with symmetric or asymmetric distributions. Finally, we provide several examples to illustrate applications of Lusin\u2019s theorem for uncertain variables. As far as we know, our results are new in uncertainty theory.<\/jats:p>","DOI":"10.3390\/sym16081034","type":"journal-article","created":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T06:02:37Z","timestamp":1723528957000},"page":"1034","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Further Results on Lusin\u2019s Theorem for Uncertain Variables"],"prefix":"10.3390","volume":"16","author":[{"given":"Deguo","family":"Yang","sequence":"first","affiliation":[{"name":"School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaojun","family":"Zong","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feng","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Qufu Normal University, Qufu 273165, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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