Skip to main content
Article
Licensed
Unlicensed Requires Authentication

An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets

  • EMAIL logo and
Published/Copyright: March 8, 2019
Become an author with De Gruyter Brill
Author Information Explore this Subject
Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 26 Issue 4

Abstract

Here we give an abstract formulation (in uncountable commutative groups) of a classical theorem of Sierpiński on the nonmeasurability of the algebraic sum of measure zero sets.

Keywords: generalized Ulam matrix; Kulikov’s theorem
MSC 2010: 28A05; 28A99; 03E05; 03E10; 28D99

References

[1] R. A. Johnson, J. Niewiarowski and T. Świa̧tkowski, Small systems convergence and metrizability, Proc. Amer. Math. Soc. 103 (1988), no. 1, 105–112. 10.1090/S0002-9939-1988-0938652-4Search in Google Scholar

[2] A. B. Kharazishvili, Some remarks on additive properties of invariant σ-ideals on the real line, Real Anal. Exchange 21 (1995/96), no. 2, 715–724. 10.2307/44152682Search in Google Scholar

[3] A. B. Kharazishvili, Nonmeasurable Sets and Functions, North-Holland Math. Stud. 195, Elsevier Science, Amsterdam, 2004. Search in Google Scholar

[4] A. B. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Stud. Math. 2, Atlantis Press, Paris, 2009. 10.2991/978-94-91216-36-7Search in Google Scholar

[5] A. B. Kharazishvili and A. Kirtadze, On algebraic sums of measure zero sets in uncountable commutative groups, Proc. A. Razmadze Math. Inst. 135 (2004), 97–103. Search in Google Scholar

[6] A. G. Kurosh, The Theory of Groups. Vol. I, Chelsea, New York, 1955. Search in Google Scholar

[7] J. Niewiarowski, Convergence of sequences of real functions with respect to small systems, Math. Slovaca 38 (1988), no. 4, 333–340. Search in Google Scholar

[8] B. Riečan, Abstract formulation of some theorems of measure theory, Mat.-Fyz. Časopis Sloven. Akad. Vied 16 (1966), 268–273. Search in Google Scholar

[9] B. Riečan, Abstract formulation of some theorems of measure theory. II, Mat. Časopis Sloven. Akad. Vied 19 (1969), 138–144. Search in Google Scholar

[10] B. Riečan, A note on measurable sets, Mat. Časopis Sloven. Akad. Vied 21 (1971), 264–268. Search in Google Scholar

[11] B. Riečan and T. Neubrunn, Integral, Measure and Ordering, Math. Appl. 411, Kluwer Academic, Dordrecht, 1997. 10.1007/978-94-015-8919-2Search in Google Scholar

[12] Z. Riécanová, On an abstract formulation of regularity, Mat. Časopis Sloven. Akad. Vied 21 (1971), 117–123. Search in Google Scholar

Received: 2017-02-02
Accepted: 2017-08-21
Published Online: 2019-03-08
Published in Print: 2019-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On absolutely Baire nonmeasurable functions
  3. Existence of special rainbow triangles in weak geometries
  4. Images of Bernstein sets via continuous functions
  5. Square-integrable representations and multipliers
  6. Semigroups of pathological sets
  7. Points of density and ideals of subsets of ℕ
  8. On Kharazishvili type measures in infinite-dimensional Polish vector spaces
  9. Families of symmetric Cantor sets from the category and measure viewpoints
  10. On the Gitik–Shelah theorem
  11. On equidistant lines of given line configurations
  12. On convergence of sequences of functions possessing closed graphs
  13. On combinatorial and set-theoretical aspects of some finite and infinite point sets
  14. On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
  15. Automorphism groups of mono-unary algebras and CH
  16. On topologies related to the extension of the Lebesgue measure
  17. An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
  18. Local affine selections of convex set-valued functions
  19. On orbits without the Baire property
  20. Mann iteration process for monotone nonexpansive mappings with a graph
  21. Mixing coded systems
  22. Methods of comparison of families of real functions in porosity terms
https://doi.org/10.1515/gmj-2019-2012
Keywords for this article
generalized Ulam matrix; Kulikov’s theorem

Articles in the same Issue

  1. Frontmatter
  2. On absolutely Baire nonmeasurable functions
  3. Existence of special rainbow triangles in weak geometries
  4. Images of Bernstein sets via continuous functions
  5. Square-integrable representations and multipliers
  6. Semigroups of pathological sets
  7. Points of density and ideals of subsets of ℕ
  8. On Kharazishvili type measures in infinite-dimensional Polish vector spaces
  9. Families of symmetric Cantor sets from the category and measure viewpoints
  10. On the Gitik–Shelah theorem
  11. On equidistant lines of given line configurations
  12. On convergence of sequences of functions possessing closed graphs
  13. On combinatorial and set-theoretical aspects of some finite and infinite point sets
  14. On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
  15. Automorphism groups of mono-unary algebras and CH
  16. On topologies related to the extension of the Lebesgue measure
  17. An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
  18. Local affine selections of convex set-valued functions
  19. On orbits without the Baire property
  20. Mann iteration process for monotone nonexpansive mappings with a graph
  21. Mixing coded systems
  22. Methods of comparison of families of real functions in porosity terms
Downloaded on 2.5.2026 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2019-2012/html?lang=en
Scroll to top button