This question was previously posted on MSE.

Let $K\subset \mathbb R^2,$ be a compact fractal of Hausdorff dimension $1<d<2.$ I want to define a natural measure on $K$.

One option would be to use the so-called Hausdorff measure $\mathcal H^d$. Where, for every $A\subset K,$ $$ \mathcal H^d(A) = \lim_{\delta\to 0} \inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\}.$$

Alternatively, given $ε >0$, one could define $K_\varepsilon =\{z\in\mathbb R^2, \mathrm{dist}(z, K) \leq \varepsilon\}$, and define $$\mu_\varepsilon(\mathrm d x) = \frac{\mathbf{1}_{K_\varepsilon} (x)}{|K_\varepsilon|} \mathrm{d}x, $$ where $\mathbf{1}_{K_\varepsilon} $ is the indicator function of $K_\varepsilon$ and $|K_\varepsilon|$ is the Lebesgue measure of $K_\varepsilon.$

Question: Is there any relation between the weak${}^*$ accumulation points of $\{\mu_\varepsilon\}_{\varepsilon >0}$ and the Hausdorff measure $\mathcal H^d$.

I could not find any book/paper that addresses this question.