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Additive functions of intervals and Hausdorff measure

Published online by Cambridge University Press:  24 October 2008

P. A. P. Moran
P. A. P. Moran
Affiliation:
St John's CollegeCambridge
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Extract

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 42 , Issue 1 , February 1946 , pp. 15 - 23
Copyright
Copyright © Cambridge Philosophical Society 1946

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References

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