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Families of surfaces lying in a null set

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Laura Wisewell
Laura Wisewell
Affiliation:
School of Mathematics, University of Edinburgh, King–s Buildings, Edinburgh, EH9 3JZ, Scotland. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT. E-mail: [email protected]
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Extract

In this note we generalize the following result of Sawyer [5]:

Theorem 1. There is a function ψ on ℝ such that, whenever g is a real-valued Borel measurable function on (a subset of) ℝ. × ℝn-1 with the property that y ↦ g(y, t) is C1 for a.e. t, the set

has measure zero.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 155 - 162
Copyright
Copyright © University College London 2004

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References

1.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge Tracts in Mathematics 85). Cambridge University Press (1985).CrossRefGoogle Scholar
2.Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces (Studies in Advanced Mathematics 44). Cambridge (1995).CrossRefGoogle Scholar
3.Mitsis, T.. (n, 2)-sets have full Hausdorff dimension. Revista Matemetica Iberoamericana. 20 (2004), 381–393.CrossRefGoogle Scholar
4.Rogers, C. A.. Hausdorff Measures. Cambridge University Press (Cambridge, 1998). (Reprintof the 1970 original, with a foreword by K. J. Falconer.)Google Scholar
5.Sawyer, E.. Families of plane curves having translates in a set of measure zero. Mathematika. 34 (1987), 69–76.CrossRefGoogle Scholar
6.Wisewell, L.. Oscillatory Integrals and Curved Kakeya Sets. Ph.D. thesis. University of Edinburgh (2003).Google Scholar
7.Wolff, T.. Recent work connected with the Kakeya problem. In Prospects in Mathematics. Princeton, New Jersey, 1996 (ed. Rossi, H.). American Mathematical Society (Providence. RI, 1999), 129–162.Google Scholar