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Transactions of the American Mathematical Society

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$ T$ measure of Cartesian product sets. II


Author: Lawrence R. Ernst
Journal: Trans. Amer. Math. Soc. 222 (1976), 211-220
MSC: Primary 28A75
DOI: https://doi.org/10.1090/S0002-9947-1976-0422587-6
MathSciNet review: 0422587
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Abstract: It is proven that there exists a subset A of Euclidean 2-space such that the 2-dimensional T measure of the Cartesian product of an interval of unit length and A is less than the 1-dimensional T measure of A. In a previous paper it was shown that there exists a subset of Euclidean 2-space such that the reverse inequality holds. T measure is the first measure of its type for which it has been shown that both of these relations are possible.


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  • [1] A. S. Besicovitch and P. A. P. Moran, The measure of product and cylinder sets, J. London Math. Soc. 20 (1945), 110-120. MR 8, 18. MR 0016448 (8:18f)
  • [2] L. R. Ernst, A proof that $ {C^2}$ and $ {T^2}$ are distinct measures, Trans. Amer. Math. Soc. 173 (1972), 501-508. MR 46 #9266. MR 0310164 (46:9266)
  • [3] -, A proof that $ {H^2}$ and $ {T^2}$ are distinct measures, Trans. Amer. Math. Soc. 191 (1974), 363-372. MR 50 #13454. MR 0361007 (50:13454)
  • [4] -, T measure of Cartesian product sets, Proc. Amer. Math. Soc. 49 (1975), 199-202. MR 0367162 (51:3404)
  • [5] L. R. Ernst and G. Freilich, A Hausdorff measure inequality, Trans. Amer. Math. Soc. 219 (1976), 361-368. MR 0419739 (54:7757)
  • [6] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. MR 41 #1976. MR 0257325 (41:1976)
  • [7] G. Freilich, On the measure of Cartesian product sets, Trans. Amer. Math. Soc. 69 (1950), 232-275. MR 12, 324. MR 0037893 (12:324a)
  • [8] -, Carathéodory measure of cylinders, Trans. Amer. Math. Soc. 114 (1965), 384-400. MR 30 #4892. MR 0174692 (30:4892)
  • [9] J. F. Randolph, On generalizations of length and area, Bull. Amer. Math. Soc. 42 (1936), 268-274. MR 1563283

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0422587-6
Keywords: 1-dimensional measures, 2-dimensional measures, Cartesian product sets, T measure, Hausdorff measure, Carathéodory measure
Article copyright: © Copyright 1976 American Mathematical Society

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