Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T13:57:06.057Z Has data issue: false hasContentIssue false
  • English
  • Français

The packing measure of rectifiable subsets of the plane

Published online by Cambridge University Press:  24 October 2008

S. James Taylor  and
Claude Tricot
S. James Taylor
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.
Claude Tricot
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Y4, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

It is usual to define Lebesgue outer measure in ℝ by using economical coverings by a sequence of open intervals. We start by outlining a different definition which gives the same answer for a bounded measurable E ⊂ ℝ. Put

Then λ0 defines a pre-measure, but is not an outer measure because it is not countably sub-additive. However it leads to an outer measure by defining

and it can be proved directly that λ is just another definition of Lebesgue outer measure. We do not give the details of this proof as it can also be deduced as a corollary of the main results in the present paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 2 , March 1986 , pp. 285 - 296
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Besicovitch, A. S.. On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann. 98 (1928), 422–464.CrossRefGoogle Scholar
[2]Besicovitch, A. S.. On the fundamental geometrical properties of linearly measurable plane sets of points II. Math. Ann. 115 (1938), 296–329.CrossRefGoogle Scholar
[3]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[4]Moore, E. F.. Density ratios and (φ, 1) rectifiability in n-space. Am. Math. Soc. Trans. 69 (1950), 324–334.Google Scholar
[5]Morse, A. P. and Randolph, J. F.. The φ-rectifiable subsets of the plane. Am. Math. Soc. Trans. 55 (1944), 236–305.Google Scholar
[6]Munroe, M.. Measure and Integration (Addison-Wesley, 1971).Google Scholar
[7]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
[8]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar