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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 $d$ ?

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  • $\begingroup$ 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. $\endgroup$ – Christian Remling Jun 8 '17 at 23:36
  • $\begingroup$ Not that I have any idea, but maybe the case of integer $d$ is simpler? $\endgroup$ – Dirk Jun 9 '17 at 2:01
  • $\begingroup$ @Dirk Or even $d=1$. $\endgroup$ – user95282 Jun 9 '17 at 11:01
  • $\begingroup$ 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. $\endgroup$ – Pablo Shmerkin Jun 10 '17 at 1:56
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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.

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