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2002/2003 The Marczewski hull property and complete boolean algebras.
Stewart Baldwin
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Real Anal. Exchange 28(2): 415-428 (2002/2003).


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.


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Stewart Baldwin. "The Marczewski hull property and complete boolean algebras.." Real Anal. Exchange 28 (2) 415 - 428, 2002/2003.


Published: 2002/2003
First available in Project Euclid: 20 July 2007

zbMATH: 1053.06007
MathSciNet: MR2009763

Primary: 28A05
Secondary: 03G05

Keywords: Complete Boolean Algebra , Marczewski Hull Property , Marczewski Measurable , Universally Measurable

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 2 • 2002/2003
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