Real Analysis Exchange

The Marczewski hull property and complete boolean algebras.

Stewart Baldwin

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Abstract

Let \(\mathcal{A}\) be an algebra of subsets of an underlying set \(A\) (which is not the entire power set of \(A\) in general), and let \({\mathcal I} \subseteq {\mathcal A}\) be an ideal over \(A\). The pair \(({\mathcal A}, {\mathcal I})\) is said to have the {\em hull property} iff whenever \(X \subseteq A\), there is a \(Y \in {\mathcal A}\) such that \(X \subseteq Y\) and \(Y\) is ``least'' mod \({\mathcal I}\), i.e., if \(Z \in {\mathcal A}\) and \(X \subseteq Z\), then \(Y \setminus Z \in {\mathcal I}\). It has been observed that in many cases for which \(({\mathcal A}, {\mathcal I})\) satisfies the hull property, the quotient Boolean algebra \({\mathcal A}/{\mathcal I}\) is a complete Boolean algebra. That, and the superficial similarity between the definitions themselves, along with the similar proofs that have sometimes resulted when using these properties, leads to the natural question of how the two properties ``\(({\mathcal A}, {\mathcal I})\) satisfies the hull property'' and ``\({\mathcal A}/{\mathcal I}\) is a complete Boolean algebra'' are related to each other. Examples will be produced which show that neither of these two properties implies the other. In addition, we examine the question of what additional hypotheses would cause one of these properties to imply the other.

Article information

Source
Real Anal. Exchange, Volume 28, Number 2 (2002), 415-428.

Dates
First available in Project Euclid: 20 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184963804

Mathematical Reviews number (MathSciNet)
MR2009763

Zentralblatt MATH identifier
1053.06007

Subjects
Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
Secondary: 03G05: Boolean algebras [See also 06Exx]

Keywords
Complete Boolean Algebra Marczewski Hull Property Marczewski Measurable Universally Measurable

Citation

Baldwin, Stewart. The Marczewski hull property and complete boolean algebras. Real Anal. Exchange 28 (2002), no. 2, 415--428. https://projecteuclid.org/euclid.rae/1184963804


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