# Naïve definition of a measure on a fractal

This question was previously posted on MSE.

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

• Thank for your answer. The only “problem” is that $K$ needs to be compact. Aug 12 at 14:10
• @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 Aug 12 at 17:48