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Transactions of the American Mathematical Society

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



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.

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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