# François G. Dorais

### Algebraic sums of sets in Marczewski–Burstin algebras

##### By F. G. Dorais and R. Filipów
###### Real Analysis Exchange 31 (2005), no. 1, 133–142

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 $A \subseteq G$ 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.