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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

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