Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the translates of a set which meet it in a set of positive measure

Author: James Foran
Journal: Proc. Amer. Math. Soc. 121 (1994), 893-895
MSC: Primary 28A78; Secondary 26A21, 28A05
MathSciNet review: 1195478
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Abstract: Given a singular Borel regular measure $ {m_a}$ on $ {R^n}$ and a Borel subset E of $ {R^n}$, it is shown that the set of vectors x for which $ {m_a}((E + x) \cap E) > 0$ is of Lebesgue measure 0. This fact is then used to show that subsets of finite, nonzero, Hausdorff s-measure are nonmeasurable sets with respect to any approximating measure $ s - {m_\delta }$.

References [Enhancements On Off] (What's this?)

  • [1] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
  • [2] Stanisław Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. MR 0167578

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Keywords: Borel regular measure, Hausdorff s-dimensional measure
Article copyright: © Copyright 1994 American Mathematical Society