TITLE: Is there a characterization of the Hausdorff measures?
QUESTION [9 upvotes]: It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue measure. 
My question is about the existence of similar result for the Hausdorff measures $H^d$.
More precisely, is there a result that says that the Hausdorff measure $H^d$ is the unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that it takes a specific value on a subset of the Borel $\sigma$-algebra (as small as possible) ? 
Edit :  to state a more precise question (but directly related)  : on $\mathbb{R}$, is the $d$-Hausdorff measure the unique translation-invariant measure that gives the value 1 to the Cantor set of dimension $d$ ?

REPLY [8 votes]: The spherical Hausdorff measure coincides with the Hausdorff measure on every rectifiable set. However it is possible to find sets where these two measures differ. See Federer 2.10.6
Hence the class of sets where one should fix the value of the measure must contain (a lot of!) non-rectifiable sets even for integer dimensions.