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On Covering Systems

Published online by Cambridge University Press:  20 November 2018

Donald J. Mallory  and
Maurice Sion
Donald J. Mallory
Affiliation:
University of British Columbia Vancouver, British Columbia
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Differentiation of a set function μ with respect to another v at a point x involves taking the limit of the ratio μA/vA as A "converges" to x in some sense which requires that A belong to some family of sets N(x) (e.g. spheres with centre x). For the development of a reasonable theory of differentiation certain restrictions must be placed on the families N(x). The best-known restriction is that they form a Vitali system. However, other systems have been considered.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 18 - 27
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Banach, S., Sur le théorème de M. Vitali, Fund. Math., 5 (1924), 134.Google Scholar
2. Hahn, H. and Rosenthal, A., Set functions (The University of New Mexico Press, 1948).Google Scholar
3. Saks, S., Theory of the integral (Warsaw, 1937).Google Scholar
4. Sion, M., Approximate continuity and differentiation, Can. J. Math., 14 (1962), 467–476.Google Scholar