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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
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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

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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