2022 Measure and Integration on Boolean Algebras of Regular Open Subsets in a Topological Space
Marcus Pivato, Vassili Vergopoulos
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Real Anal. Exchange 47(1): 25-62 (2022). DOI: 10.14321/realanalexch.47.1.1616571168


The regular open subsets of a topological space form a Boolean algebra, where the join of two regular open sets is the interior of the closure of their union. A content is a finitely additive measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such contents. We then explain the relationship between contents, residual charges, and Borel measures. We show that a content can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the measure of their common boundary. In particular, a content on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how contents can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets.


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Marcus Pivato. Vassili Vergopoulos. "Measure and Integration on Boolean Algebras of Regular Open Subsets in a Topological Space." Real Anal. Exchange 47 (1) 25 - 62, 2022. https://doi.org/10.14321/realanalexch.47.1.1616571168


Published: 2022
First available in Project Euclid: 13 June 2022

Digital Object Identifier: 10.14321/realanalexch.47.1.1616571168

Primary: 28C15 , 60B05
Secondary: 28A60

Keywords: Boolean algebra , Borel measure , compactification , Gleason cover , regular open sets , Stone space

Rights: Copyright © 2022 Michigan State University Press


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Vol.47 • No. 1 • 2022
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