A proof that ℋ² and 𝒯² are distinct measures

Ernst, Lawrence R.

doi:10.1090/S0002-9947-1974-0361007-5

A proof that $\mathcal {H}^2$ and $\mathcal {T}^2$ are distinct measures
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by Lawrence R. Ernst PDF

Trans. Amer. Math. Soc. 191 (1974), 363-372 Request permission

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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