Real Analysis Exchange

The Marczewski hull property and complete boolean algebras.

Stewart Baldwin

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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

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

First available in Project Euclid: 20 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Complete Boolean Algebra Marczewski Hull Property Marczewski Measurable Universally Measurable


Baldwin, Stewart. The Marczewski hull property and complete boolean algebras. Real Anal. Exchange 28 (2002), no. 2, 415--428.

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  • S. Baldwin, J. B. Brown, A Simple Proof that $(s)/(s^0)$ is a complete Boolean Algebra, Real Analysis Exchange, 24 (1998/9), 855–8.
  • M. Balcerzak, A. Bartoszewicz, K. Ciesielski, On Marczewski-Burstin representations of certain algebras of sets, Real Analysis Exchange, 26 (2) (2000–2001), 581–591.
  • M. Balcerzak, A. Bartoszewicz, K. Ciesielski, Algebras with inner MB-represention, preprint.
  • J. B. Brown, G. V. Cox, Classical Theory of Totally Imperfect Spaces, Real Analysis Exchange, 7 (1981-2), 185–232.
  • J. B. Brown, H. Elalaoui-Talibi, Marczewski-Burstin-like characterizations of $\sigma$-algebras, ideals, and measurable functions, Colloquium Mathematicum, 82 (1999), 277–286.
  • T. Jech, Set Theory, Academic Press, New York, San Francisco, London, 1978.
  • E. Szprilrajn (Marczewski), Sur une classe de fonction de M. Sierpinski et la classe correspondante d'ensembles, Fundamenta Mathematica, 24 (1935), 17–34.
  • A. W. Miller, S. G. Posvassilev, Vitali sets and Hamel bases that are Marczewski measurable, Fundamenta Mathematicae, 166 (2000), 269–279.
  • M. H. Stone, The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, 40 (1936), 37–111.
  • J. T. Walsh, Marczewski Sets, Measure, and the Baire Property, II, Proceedings of the American Mathematical Society, 106 (1989), 1027–1030.