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Transactions of the American Mathematical Society

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A proof that $ \mathcal{H}^2$ and $ \mathcal{T}^2$ are distinct measures

Author: Lawrence R. Ernst
Journal: Trans. Amer. Math. Soc. 191 (1974), 363-372
MSC: Primary 28A75
MathSciNet review: 0361007
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Abstract: It is proven that there exists a subset E of $ {{\mathbf{R}}^3}$ such that the two-dimensional $ \mathcal{J}$ measure of E is less than its two-dimensional Hausdorff measure. E is the image under the usual isomorphism of $ {\mathbf{R}} \times {{\mathbf{R}}^2}$ onto $ {{\mathbf{R}}^3}$ of the Cartesian product of $ \{ x: - 4 \leq x \leq 4\} $ and a Cantor type subset of $ {{\mathbf{R}}^2}$; the latter term in this product is the intersection of a decreasing sequence, every member of which is the union of certain closed circular disks.

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

  • [1] L. R. Ernst, A proof that $ {\mathcal{C}^2}$ and $ {\mathcal{J}^2}$ are distinct measures, Trans. Amer. Math. Soc. 173 (1972), 501-508. MR 0310164 (46:9266)
  • [2] H. Federer, Geometric measure theory, Die Grundlehren der math. Wissenschaften, Band 153, Springer-Verlag, New York, 1969. MR 41 #1976. MR 0257325 (41:1976)
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  • [4] E. F. Moore, Convexly generated k-dimensional measures, Proc. Amer. Math. Soc. 2 (1951), 597-606. MR 13, 218. MR 0043175 (13:218c)

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Keywords: m-dimensional measures, two-dimensional $ \mathcal{J}$ measure, two-dimensional Hausdorff measure, Cantor type subsets
Article copyright: © Copyright 1974 American Mathematical Society

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