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 measure that gives the value 1 to any Cantor set of dimension $d$ ?

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 $d$ ?

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) ?

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 measure that gives the value 1 to any Cantor set of dimension $d$ ?