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Hausdorff measure on o-minimal structures

Published online by Cambridge University Press:  12 March 2014

A. Fornasiero  and
E. Vasquez Rifo
A. Fornasiero
Affiliation:
Institut für Mathematische Logik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62. 48149 Münster, Germany, E-mail: antongiulio.fornasiero@googlemail.com
E. Vasquez Rifo
Affiliation:
Instituto de Matemática y Física, Universidad de Talca, Talca, Chile, E-mail: evasquez@inst-mat.utalca.cl
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Abstract

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy–Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

Keywords

O-minimalityHausdorff measureWhitney arc propertyCauchy-Crofton coarea
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 631 - 648
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[B004] Berarducci, A. and Otero, M.. An additive measure in o-minimal expansions of fields. The Quarterly Journal of Mathematics, vol. 55 (2004). no. 4. pp. 411–419.CrossRefGoogle Scholar
[BP98] Baisalov, Y. and Poizat, B., Paires de structures o-minirnales. this Journal, vol. 63 (1998). no. 2, pp. 570–578.Google Scholar
[Dries98] van Den Dries, L., Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press. Cambridge. 1998.Google Scholar
[Dries03] van Den Dries, L., Limit sets in o-minimal structures. O-minimal Structures, Proceedings of the RAAG Summer School Lisbon 2003 (Edmundo, M., Richardson, D., and Wilkie, A., editors). Lecture Notes in Real Algebraic and Analytic Geometry, Cuvillier Verlag. 2005.Google Scholar
[Halmos50] Halmos, P. R., Measure Theory, D. Van Nostrand Company, Inc., New York, 1950.Google Scholar
[K92] Kurdyka, K., On a subanalytic stratification satisfying a Whitney property with exponent 1. Real algebraic geometry proceedings (Rennes, 1991). Lecture Notes in Mathematics, vol. 1524, Springer, Berlin, 1992. pp. 316–322.Google Scholar
[M09] Maříková, J., 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, vol. 171 (2009), pp. 175–195.Google Scholar
[MillerO1] Miller, C., Expansions of dense linear orders with the intermediate value property, this Journal, vol. 66 (2001). no. 4. pp. 1783–1790.Google Scholar
[Morgan88] Morgan, F., Geometric measure theory, Academic Press Inc., 1988. An introduction to Federer's book by the same title.Google Scholar
[P08] Pawłucki, W., Lipschitz cell decomposition in o-minimal structures. I, Illinois Journal of Mathematics, vol. 52 (2008). no. 3, pp. 1045–1063.CrossRefGoogle Scholar
[PW06] Pila, J. and Wilkie, A. J., The rational points of a definable set, Duke Mathematical Journal. vol. 133 (2006), no. 3, pp. 591–616.Google Scholar
[VR06] Rifo, E. Vasquez, Geometric partitions of definable sets, Ph.D. thesis, The University of Wisconsin-Madison, 08 2006.Google Scholar
[W00] Warner, F., Foundations of differentiable manifolds and lie groups. Springer-Verlag, 2000.Google Scholar