Let $X$ be a compact Polish space without isolated points and $\mu$ be a finite atomless Borel measure on $X$. Then, for all $0 < \varepsilon < \mu(X)$, is there a subset $A$ of $X$ which is isomorphic to the Cantor space $\{0, 1\}^\mathbb{N}$ and has a measure $\ge \mu(X)-\varepsilon$? What if $\mu$ is equal to $s$-dimensional Hausdorff measure for some $s > 0$?
I thought the usual method that embeds the Cantor space to a Polish space can be used. But I didn't know if I could guarantee that the measure wouldn't get too small.