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

##### By F. G. Dorais and R. Filipów

###### Real Analysis Exchange 31 (2005), no. 1, 133–142

- mr: 2218194
- zbl: 1106.28001
- url: projecteuclid.org/euclid.rae/1149516801

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 is a perfect Abelian Polish group then there exists a Marczewski null set such that is not Marczewski measurable, and we show that the family of Marczewski measurable real valued functions defined on does not have the difference property.