Borel probability measure that takes value zero on all subsets with lower Hausdorff dimension

Question

Let $X\subseteq \mathbb{R}^n$ be a compact subset, assume $X$ has Hausdorff dimension $0<s<n$. I want to find condition on $X$ in order to guarantee that there exists a Probability Borel measure $\mu$ on $X$ such that $\mu(E)=0$ whenever $E\subseteq X$ has Hausdorff dimension less than $s$.