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 $K_\varepsilon =\{z\in\mathbb R^2, \operatorname{dist}(z, K) \leq \varepsilon\}$, and define $$\mu_\varepsilon(\mathrm d x) = \frac{\mathbf{1}_{K_\varepsilon} (x)}{\lvert K_\varepsilon\rvert} \mathrm{d}x, $$ where $\mathbf{1}_{K_\varepsilon} $ is the indicator function of $K_\varepsilon$ and $\lvert K_\varepsilon\rvert$ is the Lebesgue measure of $K_\varepsilon$.

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

I could not find any book/paper that addresses this question. I am particularly interested in the case where $K$ is a Julia set $J_c = \partial\{z\in\mathbb C; p_c^n(z) \not \to \infty\ \text{as }n\to\infty\},$ where $p_c(z) = z^2 +c$, for some $c$ in the Mandelbrot set.

  • $\begingroup$ Thank for your answer. The only “problem” is that $K$ needs to be compact. $\endgroup$ Aug 12 at 14:10
  • $\begingroup$ @Asaf Probably, I am not following what you are trying to say. Let us focus on the $1$-dimensional case. Do you mean to take $K= \mathbb Q \cap [0,1]$? In this case, $K$ is not compact and therefore it is not a counter-example $\endgroup$ Aug 12 at 17:48


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