On Marczewski-Burstin Representations of Certain Algebras of Sets

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.