A proof that and are distinct measures

Author:
Lawrence R. Ernst

Journal:
Trans. Amer. Math. Soc. **191** (1974), 363-372

MSC:
Primary 28A75

MathSciNet review:
0361007

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Abstract | References | Similar Articles | Additional Information

Abstract: It is proven that there exists a subset *E* of such that the two-dimensional measure of *E* is less than its two-dimensional Hausdorff measure. *E* is the image under the usual isomorphism of onto of the Cartesian product of and a Cantor type subset of ; the latter term in this product is the intersection of a decreasing sequence, every member of which is the union of certain closed circular disks.

**[1]**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**[2]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[3]**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**[4]**Edward F. Moore,*Convexly generated 𝑘-dimensional measures*, Proc. Amer. Math. Soc.**2**(1951), 597–606. MR**0043175**, 10.1090/S0002-9939-1951-0043175-8

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0361007-5

Keywords:
*m*-dimensional measures,
two-dimensional measure,
two-dimensional Hausdorff measure,
Cantor type subsets

Article copyright:
© Copyright 1974
American Mathematical Society