{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:55:33Z","timestamp":1760237733703,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Gyeongsang National University, Jinju, Korea"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonn\u00e9 measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonn\u00e9 measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined.<\/jats:p>","DOI":"10.3390\/axioms11090425","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T21:03:51Z","timestamp":1661375031000},"page":"425","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2667-1446","authenticated-orcid":false,"given":"Susmit","family":"Bagchi","sequence":"first","affiliation":[{"name":"Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660-701, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"101","DOI":"10.2140\/pjm.1968.27.101","article-title":"On supports of regular Borel measures","volume":"27","author":"Hebert","year":"1968","journal-title":"Pac. 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