example of a measure with infinite dimension of hausdorff

Question

Let $\mu$ a measure of probability non atomic of a compact metric space $X$. The dimension de $\mu$ is $$\dim \mu=\inf\{ \dim_H(Z):\mu(Z)=1\},$$ where $\dim_H(Z)$ is the hausdorff dimension of $Z\subset X$.

I wonder if you can find an example of $\mu$ such that $\dim \mu=\infty$ ?