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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

REFERENCES

(1)Gillis, J.Proc. Cambridge Phil. Soc. 33 (1937), 419–24.CrossRefGoogle Scholar
(2)Besicovitch, A. S.Math. Ann. 115 (1938).Google Scholar
(3)Besicovitch, A. S. and Moran, P. A. P. In the Press.Google Scholar
(4)Jeffrey, R. L.Trans. American Math. Soc. 35 (1933), 629–47.Google Scholar
(5)Besicovitch, A. S.Math. Ann. 101 (1929), 161–93.CrossRefGoogle Scholar