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A general form of the covering principle and relative differentiation of additive functions

Published online by Cambridge University Press:  24 October 2008

A. S. Besicovitch
A. S. Besicovitch
Affiliation:
Trinity CollegeCambridge
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Extract

The Vitali covering principle is a powerful method in a wide class of problems of the theory of functions of a real variable and of the theory of sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 41 , Issue 2 , June 1945 , pp. 103 - 110
Copyright
Copyright © Cambridge Philosophical Society 1945

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References

* Besicovitch, A. S., On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann. 115 (1938), 295–329. Lemma 2 establishes the principle for m = 1 and n = 2, but the same argument is applicable for any n and m < n.CrossRefGoogle Scholar

* We always assume that Ξ contains all the Borel sets.

* For the case of linear sets the inequality is proved with k = 1 in my paper “On Linear Sets of Points of Fractional Dimensions.” Math. Ann. 101 (1929), pp. 161–193Google Scholar. It can be generalized to the case of sets in n-dimensional space.

† We say that a family Γ of closed sets covers a set A in the sense of Vitali if to any point x of A correspond a number α = α(x) > 0 and a sequence {c n(x)} of sets of Γ satisfying the conditions

(dc n(x) is the diameter of c n(x)).