Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given $\epsilon > 0$ does there exist an open set $U \subseteq [0,1]$ such that

(i) $\lambda(A_n \setminus U) = 0$ for some $n$,

(ii) $\lambda(U) \leq \epsilon$, and

(iii) $\lambda(\partial U) = 0$,

where $\partial U$ is the boundary of $U$?

I don't have a good intuition whether this should be true or not. One idea might be to cover some $A_n$ by an open set with measure much smaller than $\epsilon$, then hope that we can expand that open set somewhat to make its boundary null.