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Packing regularity of sets in n-space

Published online by Cambridge University Press:  24 October 2008

Xavier Saint Raymond  and
Claude Tricot
Xavier Saint Raymond
Affiliation:
Département de mathématiques, Université Paris-Sud, 91405 Orsay, France
Claude Tricot
Affiliation:
Département de mathématiques, Université du Québec à Montréal, Montréal, P.Q. H3C 3P8, Canada
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Extract

The notion of packing measure, introduced in [12], [13] and [10], has been used by comparison with Hausdorif measure to study the regularity and rectifiability of sets in the plane [11]. Since a few technical mistakes can be found in [10], lemma 5·11 and [11], lemma 3·2, we wish in this paper to give the corresponding exact proofs, together with a natural development of this theory in higher dimensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 133 - 145
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Besicovitch, A. S.. On the fundamental properties of linearly measurable plane sets of points. Math. Ann. 98 (1928), 422–464.Google Scholar
[2]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).Google Scholar
[3]Federer, H.. Geometric measure theory. Grundlehren der Math. Wiss. 153 (Springer-Verlag 1969).Google Scholar
[4]de Guzman, M.. Real Variable methods in Fourier Analysis, Mathematics studies 46 (North-Holland, 1981).Google Scholar
[5]Marstrand, J. M.. The (ø,s)-regular subsets of n-space. Trans. Amer. Math. Soc. 113 (1964), 369–392.Google Scholar
[6]Mattila, P.. Hausdorff m-regular and rectifiable sets in n-space. Trans. Amer. Math. Soc. 205 (1975), 263–274.Google Scholar
[7]Moore, E. F.. Density ratios and (ø, l)-rectifiability in n-space. Trans. Amer. Math. Soc. 55 (1944), 236–305.Google Scholar
[8]Preiss, D.. Geometry of measures in Rn: distribution, rectifiability and densities, preprint.Google Scholar
[9]Rogers, C. A.. Hausdorff measures (Cambridge University Press, 1970).Google Scholar
[10]Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.CrossRefGoogle Scholar
[11]Taylor, S. J. and Tricot, C.. The packing measure of rectifiable subsets of the plane. Math. Proc. Cambridge Philos. Soc. 99 (1986), 285–296.Google Scholar
[12]Tricot, C.. Sur la classification des ensembles boréliens de mesure de Lebesgue nulle. Thèse de doctorat, Genève 1979.Google Scholar
[13]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar