ALGEBRAIC SUMS OF SETS IN MARCZEWSKI-BURSTIN ALGEBRAS.

Using almost-invariant sets, we show that a family of Marczewski-Burstin algebras over groups are not closed under algebraic sums. We also give an application of almost-invariant sets to the difference property in the sense of de Bruijn. In particular, we show that if G is a perfect Abelian Polish group then there exists a Marczewski null set A ⊆ G such that A + A is not Marczewski measurable, and we show that the family of Marczewski measurable real valued functions defined on G does not have the difference property. [ABSTRACT FROM AUTHOR]