Hausdorff $m$ regular and rectifiable sets in $n$-space
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- Trans. Amer. Math. Soc. 205 (1975), 263-274 Request permission
Abstract:
The purpose of this paper is to prove the following theorem: If $E$ is a subset of Euclidean $n$-space and if the $m$-dimensional Hausdorff density of $E$ exists and equals one ${H^m}$ almost everywhere in $E$, then $E$ is countably $({H^m},m)$ rectifiable. Here ${H^m}$ is the $m$-dimensional Hausdorff measure. The proof is a generalization of the proof given by J. M. Marstrand in the special case $n = 3,m = 2$.References
- Herbert Federer, The $(\varphi ,k)$ rectifiable subsets of $n$-space, Trans. Amer. Math. Soc. 62 (1947), 114–192. MR 22594, DOI 10.1090/S0002-9947-1947-0022594-3
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- J. M. Marstrand, Hausdorff two-dimensional measure in $3$-space, Proc. London Math. Soc. (3) 11 (1961), 91–108. MR 123670, DOI 10.1112/plms/s3-11.1.91
- Edward F. Moore, Density ratios and $(\phi ,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc. 69 (1950), 324–334. MR 37894, DOI 10.1090/S0002-9947-1950-0037894-0
- A. P. Morse and John F. Randolph, The $\phi$ rectifiable subsets of the plane, Trans. Amer. Math. Soc. 55 (1944), 236–305. MR 9975, DOI 10.1090/S0002-9947-1944-0009975-6
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 205 (1975), 263-274
- MSC: Primary 28A75
- DOI: https://doi.org/10.1090/S0002-9947-1975-0357741-4
- MathSciNet review: 0357741