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Derivative of Singular Set-Functions

Published online by Cambridge University Press:  20 November 2018

Morteza Anvari
Morteza Anvari*
Affiliation:
The University of British Columbia
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The purpose of this paper is to prove that the general derivative of a completely additive singular set-function defined on certain measurable subsets of an abstract measure space is zero almost everywhere. As a corollary the celebrated Lebesgue decomposition theorem has been sharpened.

This result is well known for set-functions defined on measurable subsets of an n-dimensional Euclidean space (2, p. 119). The proof in this setting depends on two things: Vitali's covering theorem and the fact that for every measurable set A there exists an open set O which contains A and the images of O and A under the set-function can be made arbitrarily close. Here the covering theorem is due to Trjitzinsky and the open set is replaced by an envelope, an entirely measure-theoretic concept.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 13 - 16
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Denjoy, A., Une extension du théorème de Vitali, Amer. J. Math. 73 (1951), 314–356.Google Scholar
2. Saks, S., Theory of the integral (New York, 1937).Google Scholar
3. Trjitzinsky, M. W. J.. Trjitzinsky, Théorie métrique dans les espaces où il y a une mesure, Mémorial des sciences mathématiques, 143 (1960).Google Scholar