Define the Hausdorff $d$-dimensional measure $\mathcal{H}^{d}(C)$ and the Hausdorff dimension of a subset $C$ of $\mathbb{R}$.

Set $s=\log 2 / \log 3$. Define the Cantor set $C$ and show that its Hausdorff $s$-dimensional measure is at most $1 .$

Let $\left(X_{n}\right)$ be independent Bernoulli random variables that take the values 0 and 2 , each with probability $\frac{1}{2}$. Define

$$\xi=\sum_{n=1}^{\infty} \frac{X_{n}}{3^{n}}$$

Show that $\xi$ is a random variable that takes values in the Cantor set $C$.

Let $U$ be a subset of $\mathbb{R}$ with $3^{-(k+1)} \leqslant \operatorname{diam}(U)<3^{-k}$. Show that $\mathbb{P}(\xi \in U) \leqslant 2^{-k}$ and deduce that, for any set $U \subset \mathbb{R}$, we have

$$\mathbb{P}(\xi \in U) \leqslant 2(\operatorname{diam}(U))^{s}$$

Hence, or otherwise, prove that $\mathcal{H}^{s}(C) \geqslant \frac{1}{2}$ and that the Cantor set has Hausdorff dimension $s$.