Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-07T18:40:53.100Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension prints

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers
C. A. Rogers
Affiliation:
Department of Statistical Science, University College London, Gower Street, London, WCIE 6BT
Get access

Extract

The Hausdorff dimension has been used for many years for assessing the sizes of sets in Euclidean and other metric spaces, see, for example, [1,2,5,6,8,10]. However, different sets with the same Hausdorff dimension may have very different characteristics, for example, a straight line segment in ℝ2 and the Cartesian product in ℝ2 of two suitably chosen Cantor sets in ℝ will both have Hausdorff dimension 1. In this paper we develop a measure-theoretic method of distinguishing between the sets of such pairs.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory

Information

Type
Research Article
Information
Mathematika , Volume 35 , Issue 1 , June 1988 , pp. 1 - 27
Copyright
Copyright © University College London 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Carathéodory, C.. Über das lineare Mass von Punktmengeneine Verallgemeinerung des Längenbegriffs. Nach. Ges. Wiss. Göttingen, (1914), 414–426.Google Scholar
2.Carathéodory, C.. Vorlesungen über reelle Funktionen. (Teubner, Leipzig-Berlin, 1918).Google Scholar
3.Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. 1, English Edn. (Interscience, New York, 1953).Google Scholar
4.Davies, Roy O.. Some remarks on the Kakeya problem. Proc. Camb. Phil. Soc., 59 (1971), 417–421.CrossRefGoogle Scholar
5.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
6.Hausdorff, F.. Dimension und äusseres Mass. Math. Ann., 79 (1919), 157–179.CrossRefGoogle Scholar
7.Hausdorff, F.. Mengenlehre, First Edition, 1914, reprinted (Chelsea, 1949).Google Scholar
8.Mandelbrot, B. B.. The Fractal Geometry of Nature,(Freeman, San Francisco, 1981).Google Scholar
9.Martstrand, J.M.. The dimension of Cartesian product sets. Proc. Camb. Phil. Soc., 50 (1954), 198–202.CrossRefGoogle Scholar
10.Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar