Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T11:40:30.530Z Has data issue: false hasContentIssue false

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
Affiliation:
Trinity CollegeCambridge

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
Copyright
Copyright © Cambridge Philosophical Society 1945

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* Besicovitch, A. S., On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann. 115 (1938), 295329. 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. 161193Google 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)).