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Transactions of the American Mathematical Society

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

 

 

A proof that $ \mathcal{C}^2$ and $ \mathcal{T}^2$ are distinct measures


Author: Lawrence R. Ernst
Journal: Trans. Amer. Math. Soc. 173 (1972), 501-508
MSC: Primary 28A10
DOI: https://doi.org/10.1090/S0002-9947-1972-0310164-3
MathSciNet review: 0310164
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Abstract: We prove that there exists a nonempty family $ X$ of subsets of $ {{\text{R}}^3}$ such that the two-dimensional Carathéodory measure of each member of $ X$ is less than its two-dimensional $ \mathcal{T}$ measure. Every member of $ X$ is the Cartesian product of 3 copies of a suitable Cantor type subset of $ {\text{R}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0310164-3
Keywords: $ m$-dimensional measures, two-dimensional Carathéodory measure, two-dimensional $ \mathcal{T}$ measure, Cantor type subsets, Steiner symmetrization
Article copyright: © Copyright 1972 American Mathematical Society