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HOW TO MAKE DAVIES' THEOREM VISIBLE

Published online by Cambridge University Press:  18 April 2001

MARIANNA CSÖRNYEI
MARIANNA CSÖRNYEI
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT; e-mail: [email protected]
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Abstract

We prove that for an arbitrary measurable set A ⊂ ℝ2 and a σ-finite Borel measure μ on the plane, there is a Borel set of lines L such that for each point in A, the set of directions of those lines from L containing the point is a residual set, and, moreover, μ(A) = μ({∪[lscr ] : [lscr ] ∈ L}). We show how this result may be used to characterise the sets of the plane from which an invisible set is visible. We also characterise the rectifiable sets C1, C2 for which there is a set which is visible from C1 and invisible from C2.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 33 , Issue 1 , January 2001 , pp. 59 - 66
Copyright
© The London Mathematical Society 2001

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Footnotes

Research supported by the Hungarian National Foundation for Scientific Research Grants F029768, T019476, and FKFP Grant 0192/1999.