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A property of Hausdorff measure

Published online by Cambridge University Press:  24 October 2008

Roy O. Davies
Affiliation:
University CollegeLeicester

Extract

From the fact that Hausdorff s-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result: Theorem A. If {En} is any increasing sequence of sets, thensEn → ∧sEn) as n → ∞.

Since ∧s X is denned (for every set X) as , the problem arises whether for every positive δ and every increasing sequence of sets one can prove

Now there are four possible ways of defining ; it is the lower bound of taken over coverings ΣUr of X by convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be (i) open and of diameter less than δ, (ii) closed and of diameter less than δ, (iii) open and of diameter not exceeding δ, or (iv) closed and of diameter not exceeding δ.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

REFERENCES

(1)Besicovitch, A. S.On existence of subsets of finite measure of sets of infinite measure. Indag. math. 14 (1952), 339–44.CrossRefGoogle Scholar
(2)Blaschke, W.Kreis und Kugel (Leipzig, 1916).CrossRefGoogle Scholar
(3)Saks, S.Theory of the integral (Warsaw, 1937).Google Scholar