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On the connexion between Hausdorff measures and generalized capacity

Published online by Cambridge University Press:  24 October 2008

S. J. Taylor
S. J. Taylor
Affiliation:
Cornell University and Birmingham University
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Extract

For any real function h(t) which is continuous and monotonic increasing for t > 0 with , Hausdorff (10) in 1918 denned a Carathéodory measure with respect to h(t) which has subsequently been known as Hausdorff measure. For analysing sets in Euclidean space, these measures have proved both useful and interesting. Given a real function Φ(t) which is continuous and monotonic decreasing for t > 0 with , Frostman(9) in 1935 denned capacity with respect to Φ(t). Lebesgue measure in Euclidean k-space is a special case of Hausdorff measure, and capacity with respect to Φ(t) becomes logarithmic capacity or Newtonian capacity in the cases , Φ(t)=1/t, respectively. The interrelationship between h-measure and Φ-capacity has been of interest in both directions: (i) in applications to function theory one may be able to determine whether or not a set has positive capacity by examining the h-measure for suitable h(t) (see, for example, (5)); (ii) it may be possible to determine the measure properties of a set from knowledge of its capacity (see, for example, (7) and (17)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 524 - 531
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

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