Skip to main content
Log in

On Countable Almost Invariant Partitions of G-Spaces

  • Published:
Ukrainian Mathematical Journal Aims and scope

For any σ -finite G-quasiinvariant measure μ given in a G-space, which is G-ergodic and possesses the Steinhaus property, it is shown that every nontrivial countable μ-almost G-invariant partition of the G-space has a μ-nonmeasurable member.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Sierpiński, “On the congruence of sets and their equivalence by finite decompositions,” Lucknow Univ. Stud., 20 (1954).

  2. S. Wagon, The Banach–Tarski Paradox, Cambridge Univ. Press, Cambridge (1985).

    MATH  Google Scholar 

  3. G. Vitali, Sul Problema Della Misura dei Gruppi di Punti di Una Retta, Tip. Gamberini Parmeggiani, Bologna (1905).

    MATH  Google Scholar 

  4. S. Solecki, “On sets nonmeasurable with respect to invariant measures,” Proc. Amer. Math. Soc., 119, No. 1, 115–124 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Solecki, “Measurability properties of sets of Vitali’s type,” Proc. Amer. Math. Soc., 119, No. 3, 897–902 (1993).

    MATH  MathSciNet  Google Scholar 

  6. P. Zakrzewski, “Measures on algebraic-topological structures,” Handbook of Measure Theory, Elsevier Amsterdam (2002), pp 1091–1130.

  7. P. Zakrzewski, “On nonmeasurable selectors of countable group actions,” Fund. Math., 202, No. 3, 281–294 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. B. Kharazishvili, Transformation Groups and Invariant Measures, World Sci., London–Singapore (1998).

    Book  MATH  Google Scholar 

  9. A. B. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Press, World Sci., Amsterdam; Paris (2009).

    Book  MATH  Google Scholar 

  10. W. Sierpiński, “Zermelo’s axiom and its role in set theory and analysis,” Mat. Sb., 31, Issue 1, 94–128 (1922).

    Google Scholar 

  11. A. G. Kurosh, Theory of Groups [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  12. A. Hulanicki, “Invariant extensions of the Lebesgue measure,” Fund. Math., 51, 111–115 (1962).

    MATH  MathSciNet  Google Scholar 

  13. Sh. S. Pkhakadze, “The theory of Lebesgue measure,” Proc. Razmadze Math. Inst., 25, 3–272 (1958).

    Google Scholar 

  14. A. B. Kharazishvili, “One example of an invariant measure,” Dokl. Akad. Nauk SSSR, 220, No. 1, 44–46 (1975).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 4, pp. 510–517, April, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kharazishvili, A.B. On Countable Almost Invariant Partitions of G-Spaces. Ukr Math J 66, 572–579 (2014). https://doi.org/10.1007/s11253-014-0954-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0954-3

Keywords

Navigation