Real Analysis Exchange

On Marczewski-Burstin Representations of Certain Algebras of Sets

Marek Balcerzak, Artur Bartoszewicz, and Krzysztof Ciesielski

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We show that the Generalized Continuum Hypothesis GCH (its appropriate part) implies that many natural algebras on $\mathbb{R}$, including the algebra $\mathcal{B}$ of Borel sets and the interval algebra $\Sigma$, are outer Marczewski-Burstin representable by families of non-Borel sets. Also we construct, assuming again an appropriate part of GCH, that there are algebras on $\mathbb{R}$ which are not MB-representable. We prove that some algebras (including $\mathcal{B}$ and $\Sigma$) are not inner MB-representable. We give examples of algebras which are inner and outer MB-representable, or are inner but not outer MB-representable.

Article information

Real Anal. Exchange, Volume 26, Number 2 (2000), 581-592.

First available in Project Euclid: 27 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E35: Consistency and independence results
Secondary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

Borel sets ultrafilters Continuum Hypothesis MB-representation


Balcerzak, Marek; Bartoszewicz, Artur; Ciesielski, Krzysztof. On Marczewski-Burstin Representations of Certain Algebras of Sets. Real Anal. Exchange 26 (2000), no. 2, 581--592.

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