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.
References
M. Balcerzak, A. Bartoszewicz, K. Ciesielski, On Marczewski-Burstin Representations of Certain Algebras of Sets, Real Anal. Exchange, 26(2) (2000-01), 581–591. MR1844137 1012.28002 euclid.rae/1214571351
M. Balcerzak, A. Bartoszewicz, K. Ciesielski, On Marczewski-Burstin Representations of Certain Algebras of Sets, Real Anal. Exchange, 26(2) (2000-01), 581–591. MR1844137 1012.28002 euclid.rae/1214571351
M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wroński, Marczewski Fields and Ideals, Real Anal. Exchange, 26(2) (2000-01), 703–715. MR1844387 1009.28001 euclid.rae/1214571361
M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wroński, Marczewski Fields and Ideals, Real Anal. Exchange, 26(2) (2000-01), 703–715. MR1844387 1009.28001 euclid.rae/1214571361
K. Ciesielski, H. Fejzić, C. Freiling, Measure Zero Sets with Non-Measurable Sum, Real Anal. Exchange, 27(2) (2001-02), 783–793. MR1923168 1048.28001 euclid.rae/1212412875
K. Ciesielski, H. Fejzić, C. Freiling, Measure Zero Sets with Non-Measurable Sum, Real Anal. Exchange, 27(2) (2001-02), 783–793. MR1923168 1048.28001 euclid.rae/1212412875
R. Filipów, On the Difference Property of Families of Measurable Functions, Colloq. Math., 97(2) (2003), 169–180. MR2031845 1067.28004 10.4064/cm97-2-4 R. Filipów, On the Difference Property of Families of Measurable Functions, Colloq. Math., 97(2) (2003), 169–180. MR2031845 1067.28004 10.4064/cm97-2-4
R. Filipów and I. Recław, On the Difference Property of Borel Measurable and $(s)$-Measurable Functions, Acta Math. Hungar., 96(1,2) (2002), 21–25. MR1912487 1006.28006 10.1023/A:1015661511337 R. Filipów and I. Recław, On the Difference Property of Borel Measurable and $(s)$-Measurable Functions, Acta Math. Hungar., 96(1,2) (2002), 21–25. MR1912487 1006.28006 10.1023/A:1015661511337
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. MR1321597 0819.04002 A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. MR1321597 0819.04002
A. B. Kharazishvili, Some Remarks on Additive Properties of Invariant $\sigma$-Ideals on the Real Line, Real Anal. Exchange, 21(2) (1995-96), 715–724. MR1407284 0879.28026 A. B. Kharazishvili, Some Remarks on Additive Properties of Invariant $\sigma$-Ideals on the Real Line, Real Anal. Exchange, 21(2) (1995-96), 715–724. MR1407284 0879.28026
M. Kysiak, Nonmeasurable Algebraic Sums of Sets of Reals, Colloq. Math., 102(1) (2005), 113-122. MR2150273 1072.28002 10.4064/cm102-1-10 M. Kysiak, Nonmeasurable Algebraic Sums of Sets of Reals, Colloq. Math., 102(1) (2005), 113-122. MR2150273 1072.28002 10.4064/cm102-1-10
M. Laczkovich, The Difference Property, Paul Erdős and his mathematics, I, Budapest, (1999), Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, (2002), 363–410. MR1954704 M. Laczkovich, The Difference Property, Paul Erdős and his mathematics, I, Budapest, (1999), Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, (2002), 363–410. MR1954704
O. Spinas, Ramsey and Freeness Properties of Polish Planes, Proc. London Math. Soc., (3), 82(1) (2001), 31–63. MR1794256 1020.03042 10.1112/S0024611500012740 O. Spinas, Ramsey and Freeness Properties of Polish Planes, Proc. London Math. Soc., (3), 82(1) (2001), 31–63. MR1794256 1020.03042 10.1112/S0024611500012740
