Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. Thus if $a$ is a power of $3$, $\mu(aC) = a^{\log_3 2} \mu(C)$.

**Question 1**: Is it the case for all $a\in (0,\infty)$ that $\mu(aC)=a^{\log_3 2} \mu(C)$, if $\mu$ is a finitely-additive isometrically-invariant measure on $\mathcal P\mathbb R$?

The remaining questions are predicated on an affirmative answer, though currently I'm suspecting a negative answer to 1.

**Question 2**: Is this still true if isometric-invariance is replaced by translation-invariance?

**Question 3**: Has there been any work on using the the relationship between $\mu(aC)$ and $\mu(C)$ for invariant measures $\mu$ on $\mathbb R^n$ (perhaps not defined on all of the powerset, but just on the Borel sets) to define a dimension and comparing it to Hausdorff and box dimensions?