Does $Leb(S_i) \rightarrow 0$ imply that $\mu(S_i) \rightarrow 0$ for a non-atomic measure $\mu \in \mathcal{P}(S^1)$?

Question

Let $\mu$ be a non-atomic probability measure on $\mathbb{R} / \mathbb{Z}$ and let m stand for Lebesgue measure on $\mathbb{R} / \mathbb{Z}$. Is it true that if $S_i \subset \mathbb{R} / \mathbb{Z}$ is a sequence of subsets which are $m$ and $\mu$-measurable and $m(S_i) \rightarrow0$ as $i \uparrow \infty$ then also $\mu(S_i) \rightarrow 0$?