𝒯 measure of Cartesian product sets

Ernst, Lawrence R.

doi:10.1090/S0002-9939-1975-0367162-1

$\mathcal {T}$ measure of Cartesian product sets
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Proc. Amer. Math. Soc. 49 (1975), 199-202 Request permission

Abstract:

It is proven that there exists a subset $A$ of Euclidean $2$-space such that the $2$-dimensional $\mathcal {T}$ measure of the Cartesian product of an interval of unit length and $A$ is greater than the $1$-dimensional $\mathcal {T}$ measure of $A$. This shows that $\mathcal {T}$ measure does not extend to Euclidean $3$-space the relation that area is the product of length by length. As corollaries, new proofs of some related but previously known results are obtained.

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