# Is there a characterization of the Hausdorff measures?

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 $d$ ?

• I very much doubt that there will be a "small" collection of such sets with "natural" values, whatever that means, unless $d=0$ or $n$. It works fine of course when restricted to suitable sets, for example the Cantor measure is the unique measure that gives each of the $2^n$ parts of the Cantor set (in the natural construction) its fair share of measure. – Christian Remling Jun 8 '17 at 23:36
• Not that I have any idea, but maybe the case of integer $d$ is simpler? – Dirk Jun 9 '17 at 2:01
• @Dirk Or even $d=1$. – user95282 Jun 9 '17 at 11:01
• The answer to the precise question is definitely no - the Cantor set of dimension $d$ has positive and finite $d$-dimensional packing measure, so a multiple of $d$-packing measure is a translation invariant measure that agrees with Hausdorff measure on the Cantor set yet is a (very) different measure. – Pablo Shmerkin Jun 10 '17 at 1:56