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

Abstract

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.

Citation

Download Citation

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

Information

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

Digital Object Identifier: 10.14321/realanalexch.47.1.1616571168

Subjects:
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

JOURNAL ARTICLE
38 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.47 • No. 1 • 2022
Back to Top