A proof that ℋ² and 𝒯² are distinct measures

Abstract

It is proven that there exists a subset E of R 3 {{\mathbf {R}}^3} such that the two-dimensional J \mathcal {J} measure of E is less than its two-dimensional Hausdorff measure. E is the image under the usual isomorphism of R × R 2 {\mathbf {R}} \times {{\mathbf {R}}^2} onto R 3 {{\mathbf {R}}^3} of the Cartesian product of { x : − 4 ≤ x ≤ 4 } \{ x: - 4 \leq x \leq 4\} and a Cantor type subset of R 2 {{\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.