Hausdorff Measure Given the Hausdorff Measure
Is it true that $H^1$(line)=  Length of the line?
How can one prove it?
 A: 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.
A: Here I explain Evans and Gariep's proof. In dimension 1 the isodiametric inequality is way easier to prove than in bigger dimensions.
https://www.youtube.com/watch?v=n4TMCzR0biM
