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A measureless one-dimensional set

Published online by Cambridge University Press:  24 October 2008

H. G. Eggleston
H. G. Eggleston
Affiliation:
7 Hauxton RoadTrumpingtonCambridge
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Extract

It has been known for some time that there are sets in Euclidean space which are of infinite measure in a certain Hausdorff dimension, say α, and yet which contain no subsets that are of finite positive a measure. The properties of such a set, say X, could be developed in much the same way as those of sets which are of positive finite α measure, if there existed a Caratheodory outer measure Γ, defined over the subsets of X, which was such that ∞ > Γ(X) > 0, and for any subset Y of X, Λα (Y) = 0 implied Γ (Y) = O. The object of this note is to show that there are sets for which no such outer measure exists. It is shown that a set defined by Sierpinski (7) is one-dimensional and is such that any Caratheodory outer measure defined over its subsets either takes infinite values or is identically zero or is not zero for all subsets that consist of a single point. It has been remarked by Prof. Besicovitch that these properties imply that the set is measurable with respect to any Caratheodory outer measure defined over subsets of the plane, and in fact any subset is measurable with respect to any such outer measure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 3 , July 1954 , pp. 391 - 393
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1)Besicovitch, A. S.Concentrated and rarefied sets of points. Acta Math. 62 (1933), 289–300.Google Scholar
(2)Besicovitch, A. S.Relations between concentrated sets and sets possessing property C. Proc. Camb. phil. Soc. 38 (1942), 20–3.Google Scholar
(3)Besicovitch, A. S.On existence of subsets of finite measure of sets of infinite measure. Indag. Math. 14 (1952), 339–44.CrossRefGoogle Scholar
(4)Davies, R. O.Subsets of finite measure in analytic sets. Indag. Math. 14 (1952), 488–9.Google Scholar
(5)Eggleston, H. G.The Besicovitch dimension of Cartesian product sets. Proc. Camb. phil. Soc. 46 (1950), 383–6.Google Scholar
Eggleston, H. G.A correction to a paper on the dimension of Cartesian product sets. Proc. Camb. phil. Soc. 49 (1953), 437–40.CrossRefGoogle Scholar
(6)Marstrand, J. M.The dimension of Cartesian product sets. Proc. Camb. phil. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
(7)Sierpinski, W.Sur le produit combinatoire de deux ensembles jouissant de la propriéte C. Fundam. Math. 24 (1935), 48–50.CrossRefGoogle Scholar
(8)Sierpinski, W.Sur un ensemble non–dénombrables dont toute image continue est de mesure nulle. Fundam. Math. 11 (1928), 302–4.CrossRefGoogle Scholar