Real Analysis Exchange

Algebraic sums of sets in Marczewski-Burstin algebras.

Francois G. Dorais and Rafał Filipów

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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.

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 133-142.

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 39A70: Difference operators [See also 47B39]

algebraic sum Marczewski--Burstin algebra Marczewski measurable set Miller measurable set perfect set superperfect set almost-invariant set difference property


Dorais, Francois G.; Filipów, Rafał. Algebraic sums of sets in Marczewski-Burstin algebras. Real Anal. Exchange 31 (2005), no. 1, 133--142.

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  • M. Balcerzak, A. Bartoszewicz, K. Ciesielski, On Marczewski-Burstin Representations of Certain Algebras of Sets, Real Anal. Exchange, 26(2) (2000-01), 581–591.
  • M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wroński, Marczewski Fields and Ideals, Real Anal. Exchange, 26(2) (2000-01), 703–715.
  • K. Ciesielski, H. Fejzić, C. Freiling, Measure Zero Sets with Non-Measurable Sum, Real Anal. Exchange, 27(2) (2001-02), 783–793.
  • N. Govert de Bruijn, Functions Whose Differences Belong to a Given Class, Nieuw Arch. Wiskunde, 23(2) (1951), 194–218.
  • R. Filipów, On the Difference Property of Families of Measurable Functions, Colloq. Math., 97(2) (2003), 169–180.
  • 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.
  • F. Galvin, Partitions Theorems for the Real Line, Notices Amer. Math. Soc., 15 (1968), 660, Erratum, 16 (1969), 1095.
  • A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.
  • 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.
  • S. Kurepa, Convex Functions, Glasnik Mat.-Fiz. Astr. Ser. II., 11 (1956), 89–94.
  • M. Kysiak, Nonmeasurable Algebraic Sums of Sets of Reals, Colloq. Math., 102(1) (2005), 113-122.
  • 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.
  • E. Marczewski (Spilrajn), Sur une Classe de Fonctions de M. Sierpiński et la Classe Correspondante d'Ensembles, Fund. Math., 24 (1935), 17–34.
  • J. Mycielski, Independent Sets in Topological Algebras, Fund. Math., 55 (1964), 139–147.
  • W. Sierpiński, Sur la Question de la Mesurabilité de la Base de M. Hamel, Fund. Math., 1 (1920), 105–111.
  • W. Sierpiński, Sur les Translations des Ensembles Linéaires, Fund. Math., 19 (1932), 22–28.
  • O. Spinas, Ramsey and Freeness Properties of Polish Planes, Proc. London Math. Soc., (3), 82(1) (2001), 31–63.