SpringerLink
Log in

On ordinary and standard products of infinite family of σ-finite measures and some of their applications

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on ℝ∞ and Rogers-Fremlin measures on ℓ ∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal (i.e., 2c). We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kuratowski, K., Mostowski, A.: Set Theory (in Russian), Nauka, Moscow, 1980

    Google Scholar 

  2. Kharazishvili, A. B.: Elements of Combinatorical Theory of Infinite Sets (in Russian), Tbilisi University Press, Tbilisi, 1981

    Google Scholar 

  3. Kharazishvili, A. B.: Invariant Extensions of the Lebesgue Measure (in Russian), Tbilis. Gos. Univ., Tbilisi, 1983

    Google Scholar 

  4. Szpilrajn, E.: Ensembles indépendants et mesures non séparables. Comptes Rendus., 207, 768–770 (1938)

    Google Scholar 

  5. Kakutani, S., Oxtoby, J.: Construction of non-separable invariant extension of the Lebesgue measure space. Ann. of Math., 52(2), 580–590 (1950)

    Article  MathSciNet  Google Scholar 

  6. Kodaira, K., Kakutani, S.: A nonseparable tranlation-invariant extension of the Lebesgue measure space. Ann. of Math., 52, 574–579 (1950)

    Article  MathSciNet  Google Scholar 

  7. Pantsulaia, G. R.: An application of independent families of sets to the measure extension problem. Georgian Math. J., 11(2), 379–390 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Loeb, P. A., Ross, D. A.: Infinite products of infinite measures. Illinois J. Math., 49(1), 153–158 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Baker, R.: “Lebesgue measure” on ℝ∞. Proc. Amer. Math. Soc., 113(4), 1023–1029 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Baker, R.: “Lebesgue measure” on ℝ∞. II. Proc. Amer. Math. Soc., 132(9), 2577–2591 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Pantsulaia, G. R.: On ordinary and standard Lebesgue Measures on ℝ∞. Bull. Pol. Acad. Sci. Math., 73(3), 209–222 (2009)

    Article  MathSciNet  Google Scholar 

  12. Fremlin, D. H.: On a Question of C. A. Rogers, University of Essex, Colchester, England, version of 17.03. 1998

  13. Bruckner, A. M., Bruckner, J. B., Thomson B. S.: Real Analysis, http://www.classicalrealanalysis.com, 2008

  14. Halmos, P. R.: Measure Theory, Van Nostrand, Princeton, 1950

    MATH  Google Scholar 

  15. Kharazishvili, A. B.: On cardinalities of isodyne topological spaces. Bull. Georgian Acad. Sci., 137(1), 33–36 (1990)

    MATH  MathSciNet  Google Scholar 

  16. Pantsulaia, G. R.: Relations between shy sets and sets of ν p-measure zero in Solovay’s model. Bull. Pol. Acad. Sci. Math., 52(1), 63–69 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Abraham, U., Shelah, S.: Coding with ladders a well ordering of the reals. J. Symbolic Logic, 67(2), 579–597 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Oxtoby, J. C.: Measure and Category, A survey of the analogies between topological and measure spaces, Second edition, Graduate Texts in Mathematics, 2, Springer-Verlag, New York-Berlin, 1980

    MATH  Google Scholar 

  19. Pantsulaia, G. R.: Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Science Publishers, New York, 2007

    MATH  Google Scholar 

  20. Pantsulaia, G. R.: On generators of shy sets on Polish topological vector spaces. New York J. Math., 14, 235–261 (2008)

    MATH  MathSciNet  Google Scholar 

  21. Christensen, J. P. R.: Uniform measures. Proc. of Functional Analysis Week (Aarhus, 1969), Matematisk Inst., Aarhus Univ., Aarhus, 8–16 (1969)

  22. Wen, S. Y.: A certain regular property of the method I construction and packing measure. Acta Mathematica Sinica, English Series, 23(10), 1769–1776 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Georgian Technical University, Faculty of Informatics and Control Systems, B. P. 0175, Kostava Street-77, Tbilisi, Georgia, USA

    Gogi Rauli Pantsulaia

  2. I. Vekua Institute of Applied Mathematics, Tbilisi State University, Division of Applied Logic and Programming B. P. 0143, University St. 2, Tbilisi, Georgia, USA

    Gogi Rauli Pantsulaia

Authors
  1. Gogi Rauli Pantsulaia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gogi Rauli Pantsulaia.

Additional information

Supported by National Science Foundation of Georgia (Grants Nos. GNSF/ST 08/3-391, Sh. Rustaveli GNSF/ST 09 144-3-105)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pantsulaia, G.R. On ordinary and standard products of infinite family of σ-finite measures and some of their applications. Acta. Math. Sin.-English Ser. 27, 477–496 (2011). https://doi.org/10.1007/s10114-011-9515-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-011-9515-y

Keywords

MR(2000) Subject Classification

Search

Navigation