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 of Euclidean -space such that the -dimensional measure of the Cartesian product of an interval of unit length and is greater than the -dimensional measure of . This shows that measure does not extend to Euclidean -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.

**[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****[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**, 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**, 10.1090/S0002-9947-1974-0361007-5**[4]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[5]**Gerald Freilich,*On the measure of Cartesian product sets*, Trans. Amer. Math. Soc.**69**(1950), 232–275. MR**0037893**, 10.1090/S0002-9947-1950-0037893-9**[6]**Gerald Freilich,*Carathéodory measure of cylinders*, Trans. Amer. Math. Soc.**114**(1965), 384–400. MR**0174692**, 10.1090/S0002-9947-1965-0174692-2**[7]**J. F. Randolph,*On generalizations of length and area*, Bull. Amer. Math. Soc.**42**(1936), no. 4, 268–274. MR**1563283**, 10.1090/S0002-9904-1936-06287-7

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1975-0367162-1

Keywords:
-dimensional measures,
-dimensional measures,
Cartesian product sets,
measure,
Hausdorff measure,
spherical measure,
Carathéodory measure

Article copyright:
© Copyright 1975
American Mathematical Society