{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:53:17Z","timestamp":1760151197231,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,20]],"date-time":"2022-02-20T00:00:00Z","timestamp":1645315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of k-finite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions.<\/jats:p>","DOI":"10.3390\/sym14020422","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T08:23:29Z","timestamp":1645431809000},"page":"422","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes"],"prefix":"10.3390","volume":"14","author":[{"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":[[2022,2,20]]},"reference":[{"key":"ref_1","first-page":"168","article-title":"The interplay between measure theory and topology","volume":"4","author":"Ash","year":"1972","journal-title":"Meas. Integr. Funct. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s10474-010-9223-6","article-title":"Borel sets and functions in topological spaces","volume":"129","author":"Spurny","year":"2010","journal-title":"Acta Math. Hung."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1090\/S0002-9904-1972-12936-7","article-title":"On the nonseparable theory of Borel and Souslin sets","volume":"78","author":"Hansell","year":"1972","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/0016-660X(72)90010-4","article-title":"Non-separable Borel sets II","volume":"2","author":"Stone","year":"1972","journal-title":"Gen. Topol. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Mashford, J. (2020). A spectral calculus for Lorentz invariant measures on Minkowski space. Symmetry, 12.","DOI":"10.3390\/sym12101696"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1090\/S0002-9947-1975-0369648-7","article-title":"A generalized topological measure theory","volume":"207","author":"Kirk","year":"1975","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"565","DOI":"10.2140\/pjm.1979.82.565","article-title":"Topological measure theory for completely regular spaces and their projective covers","volume":"82","author":"Wheeler","year":"1979","journal-title":"Pac. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1006\/aima.1999.1866","article-title":"Borel measure extensions of measures defined on sub-\u03c3-algebras","volume":"150","author":"Aldaz","year":"2000","journal-title":"Adv. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Butler, S.V. (2021). Weak Convergence in Topological Measures. J. Theor. Probab.","DOI":"10.1007\/s10959-021-01095-4"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1090\/S0002-9939-98-04107-0","article-title":"The Orlicz-Pettis theorem for topological Reisz spaces","volume":"126","author":"Drewnowski","year":"1998","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.topol.2004.09.007","article-title":"Extending Baire-one functions on topological spaces","volume":"149","author":"Kalenda","year":"2005","journal-title":"Topol. Its Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1007\/s10958-018-3923-9","article-title":"Semiring of continuous (0,\u221e]-valued functions","volume":"233","author":"Vechtomov","year":"2018","journal-title":"J. Math. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.4064\/sm181107-16-2","article-title":"Representation of increasing convex functionals with countably additive measures","volume":"260","author":"Cheridito","year":"2021","journal-title":"Studia Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1770","DOI":"10.1007\/s12220-013-9394-9","article-title":"Measures and dynamics on Noetherian spaces","volume":"24","author":"Gignac","year":"2014","journal-title":"J. Geom. 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