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 measure of Cartesian product sets. II
Author:
Lawrence R. Ernst
Journal:
Trans. Amer. Math. Soc. 222 (1976), 211-220
MSC:
Primary 28A75
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.
- [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 0016448
(8,18f)
- [2]
Lawrence
R. Ernst, A proof that
∗∗&𝑐𝑠𝑐𝑟𝐶∗∗𝑐𝑠𝑐𝑟𝐶²
and
∗∗&𝑐𝑠𝑐𝑟𝑇∗∗𝑐𝑠𝑐𝑟𝑇²
are distinct measures, Trans. Amer. Math.
Soc. 173 (1972),
501–508. MR 0310164
(46 #9266), http://dx.doi.org/10.1090/S0002-9947-1972-0310164-3
- [3]
Lawrence
R. Ernst, A proof that
\𝑐𝑎𝑙𝐻² and
\𝑐𝑎𝑙𝑇² are distinct measures,
Trans. Amer. Math. Soc. 191 (1974), 363–372. MR 0361007
(50 #13454), http://dx.doi.org/10.1090/S0002-9947-1974-0361007-5
- [4]
Lawrence
R. Ernst, 𝒯 measure of Cartesian product
sets, Proc. Amer. Math. Soc. 49 (1975), 199–202. MR 0367162
(51 #3404), http://dx.doi.org/10.1090/S0002-9939-1975-0367162-1
- [5]
Lawrence
R. Ernst and Gerald
Freilich, A Hausdorff measure
inequality, Trans. Amer. Math. Soc. 219 (1976), 361–368. MR 0419739
(54 #7757), http://dx.doi.org/10.1090/S0002-9947-1976-0419739-8
- [6]
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
- [7]
Gerald
Freilich, On the measure of Cartesian product
sets, Trans. Amer. Math. Soc. 69 (1950), 232–275. MR 0037893
(12,324a), http://dx.doi.org/10.1090/S0002-9947-1950-0037893-9
- [8]
Gerald
Freilich, Carathéodory measure of
cylinders, Trans. Amer. Math. Soc. 114 (1965), 384–400. MR 0174692
(30 #4892), http://dx.doi.org/10.1090/S0002-9947-1965-0174692-2
- [9]
J.
F. Randolph, On generalizations of length and
area, Bull. Amer. Math. Soc.
42 (1936), no. 4,
268–274. MR
1563283, http://dx.doi.org/10.1090/S0002-9904-1936-06287-7
- [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
 and  are distinct measures, Trans. Amer. Math. Soc. 173 (1972), 501-508. MR 46 #9266. MR 0310164 (46:9266)
- [3]
- -, A proof that
 and  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:
http://dx.doi.org/10.1090/S0002-9947-1976-0422587-6
PII:
S 0002-9947(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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