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A note on Hausdorff dimension functions

Published online by Cambridge University Press:  24 October 2008

A. Dvoretzky
A. Dvoretzky
Affiliation:
Hebrew UniversityJerusalem
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Extract

1. We recall the meaning attributed by F. Hausdorff to the statement: ‘The (linear) set S is of dimension [h(x)].’

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 1 , January 1948 , pp. 13 - 16
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

* Hausdorff, F., Math. Ann. 79 (1918), 157–79,CrossRefGoogle Scholar especially pp. 167–77. The basic ideas are due to Caratheodory, C., Gött. Nachr. 1914, pp. 404–26.Google Scholar

† Prof. M. Fekete's first results in this direction were found in 1936–7 when he held in the Hebrew University a seminar on the theory of measure. He summarized all results found then and subsequently in a lecture delivered in 1945 in the Mathematical Society of Jerusalem, where he also posed the problem, solved in the present note, of characterizing measure functions which are also dimension functions. The related problem whether to every bounded (even closed) linear set S there corresponds a measure function h(x) such that S is of dimension [h(x)] has been answered negatively by Best, E., ‘A closed dimensionless linear set’, Proc. Edinburgh Math. Soc. (2) 6 (1939), 105–8.CrossRefGoogle Scholar

* The fact that S was covered by the closed intervals J n is immaterial, since h(x) is continuous.

* See footnote *, p. 13.