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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Gross measure property

Author: Lawrence R. Ernst
Journal: Trans. Amer. Math. Soc. 238 (1978), 397-406
MSC: Primary 28A75
MathSciNet review: 0476999
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Abstract: We prove that there exists a subset E of $ [0,1] \times {{\mathbf{R}}^2}$ such that the 2-dimensional Gross measure of E is 0, while the 1-dimensional Gross measure of $ \{ z:(y,z) \in E\} $ is positive for all $ y \in [0,1]$. It is known that for Hausdorff measures no set exists satisfying these conditions.

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

  • [1] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [2] D. J. Ward, A counter-example in area theory, Proc. Cambridge Philos. Soc. 60 (1964), 821–845. MR 170997
  • [3] Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148

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Keywords: Gross measure, Hausdorff measure, structure theory, $ ({\mathcal{G}^m},m)$ rectifiable
Article copyright: © Copyright 1978 American Mathematical Society