We show that the Generalized Continuum Hypothesis GCH (its appropriate part)
implies that many natural algebras on **R**,
including the algebra **B** of Borel sets and the interval algebra **S**,
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 **R** which are not
MB-representable. We prove that some algebras
(including **B** and **S**) 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.

**Proof of Theorem 7 from the paper has an error!**
We do not know if the theorem is true or false.

See also related paper:

- M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wronski,
Marczewski fields and ideals,
*Real Anal. Exchange 26(2)*(2000-2001), 703-715.

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**Last modified November 16, 2003.**