Skip to main content
SpringerLink
Log in

Measuring definable sets in o-minimal fields

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce a non-real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field R. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of R. We show that every measurable subset of R n with non-empty interior has positive measure, and that the measure is preserved by definable C 1-diffeomorphisms with Jacobian determinant equal to ±1.

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

Access this article

Log in via an institution

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. A. Berarducci and M. Otero, An additive measure in o-minimal expansions of fields, The Quarterly Journal of Mathematics 55 (2004), 411–419.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. van den Dries, O-minimal structures, in Logic: from Foundations to Applications (Staffordshire, 1993), Oxford Science Publications, Oxford University Press, New York, 1996, pp. 137–185.

    Google Scholar 

  3. L. van den Dries, Tame Topology and o-minimal Structures, London Mathematical Society Lecture Note Series, Vol. 248, Cambridge University Press, Cambridge, 1998.

    Book  MATH  Google Scholar 

  4. C. F. Ealy and J. Maříková, Model completeness of o-minimal fields with convex valuations, Journal of Symbolic Logic, to appear, (arxiv.org/abs/1211.6755).

  5. E. Hrushovski and Y. Peterzil, A question of van den Dries and a theorem of Lipshitz and Robinson; not everything is standard, Journal of Symbolic Logic 72 (2007), 119–122.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Hrushovski, Y. Peterzil and A. Pillay, Groups, measures and the NIP, Journal of the American Mathematical Society 21 (2008), 563–596.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. Lipshitz and Z. Robinson, Overconvergent real closed quantifier elimination, Bulletin of the London Mathematical Society 38 (2006), 897–906.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Maříková, The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field, Israel Journal of Mathematics 171 (2009), 175–195.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Maříková, O-minimal fields with standard part map, Fundamenta Mathematicae 209 (2010), 115–132.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Shiota, Geometry of Subanalytic and Semialgebraic Sets, Progress in Mathematics, Vol. 150, Birkhäuser, Boston, MA, 1997.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Western Illinois University, 476 Morgan Hall, 1 University Circle, Macomb, IL, 61455, USA

    Jana Maříková

Authors
  1. Jana Maříková
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masahiro Shiota
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jana Maříková.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maříková, J., Shiota, M. Measuring definable sets in o-minimal fields. Isr. J. Math. 209, 687–714 (2015). https://doi.org/10.1007/s11856-015-1234-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-015-1234-0

Keywords

Search

Navigation