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Functions continuous and singular with respect to a Hausdorff measure

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers  and
S. J. Taylor
C. A. Rogers
Affiliation:
University College, London
S. J. Taylor
Affiliation:
Cornell University, and Birmingham University
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Extract

1. Introduction. Let I0 be a closed rectangle in Euclidean n-space, and let ℬ be the field of Borel subsets of I0. Let ℱ be the space of completely additive set functions F, having a finite real value F(E) for each E of ℬ, and left undefined for sets E not in ℬ. In recent work, we used Hausdorff measures in an attempt to analyze the set functions F of ℱ. If h(t) is a monotonic increasing continuous function of t with h(0) = 0, a measure h-m(E) is generated by the method first defined by Hausdorff [2].

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Type
Research Article
Information
Mathematika , Volume 8 , Issue 1 , June 1961 , pp. 1 - 31
Copyright
Copyright © University College London 1961

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References

1.Besicovitch, A. S., “On the definition of tangents to sets of infinite linear measure”, Proc. Camb. Phil. Soc., 52 (1956), 20–29.CrossRefGoogle Scholar
2.Hausdorff, F., “Dimension und ausseres Mass”, Math. Ann., 79 (1919), 157–179.CrossRefGoogle Scholar
3.Mickle, E. J. and Rado, T., “On reduced Cartheodory outer measures”, Rendi. del Circ. Mat. di Palermo, (2), 7 (1958), 5–33.Google Scholar
4.Rogers, C. A. and Taylor, S. J., “The analysis of additive set functions in Euclidean space”, Acta Math., 101 (1959), 273–302.CrossRefGoogle Scholar
5.Rogers, C. A. and Taylor, S. J., “The analysis of additive set functions in Euclidean space, II”. In preparation.Google Scholar
6.Saks, S., Theory of the Integral. Warsaw, 1937.Google Scholar