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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \mathcal{T}$ measure of Cartesian product sets


Author: Lawrence R. Ernst
Journal: Proc. Amer. Math. Soc. 49 (1975), 199-202
MSC: Primary 28A75
MathSciNet review: 0367162
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0367162-1
PII: S 0002-9939(1975)0367162-1
Keywords: $ 1$-dimensional measures, $ 2$-dimensional measures, Cartesian product sets, $ \mathcal{T}$ measure, Hausdorff measure, spherical measure, Carathéodory measure
Article copyright: © Copyright 1975 American Mathematical Society