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Given the Hausdorff Measure

Is it true that $H^1$(line)= Length of the line?

How can one prove it?

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What exactly do you mean by a line? – copper.hat Jul 22 '12 at 6:18
By line do you mean a subset of $\mathbb{R}^{2}$ or an interval i.e. a subset of $\mathbb{R}$? The answer is different depending on what you mean. – T. Eskin Jul 22 '12 at 8:08
up vote 3 down vote accepted

On $\mathbb{R}^n$, thus in particular on $\mathbb{R}$, the $n-$ dimensional Hausdorff measure equals the $n-$ dimensional Lebesgue measure. This is not completely trivial for general $n$, not too difficult for $n=1$. One source for this is L.C.Evans, R.F.Gariepy, 'Measure theory and fine properties of functions'. The equality of the measures is Theorem 2 in chapter 2.3. The proof for $n= 1$ is in section 2.1, Theorem 2 (or, of course, Federer's monograph :-))

Edit: another source is William P. Ziemer, 'Weakly differentiable functions', Theorem 1.4.2.

another edit: the wikipedia page you used as a reference uses a definition of Hausdorff measure which differs by a dimension dependend factor from the one in the sources I cited. Consequently the Hausdorff measure, using that definition, is a factor times the Lebesgue measure. This is also stated on that page, and the factor is explicitly given.

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Thanks a lot !!!! – joshua Jul 22 '12 at 7:04

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