Open set containing cantor set

Question

Is it possible to construct an open set of measure less than a given positive real and containing cantor set?

Answer 1

Yes. The middle-thirds cantor set is constructed as the intersection of nested closed sets $J_n $ of decreasing measure. Each of these closed sets is the union of finitely many closed intervals $I_{n_1},...,I_{n_m}$. Given $\epsilon>0$, find $n $ sufficiently large that the measure of $J_n $ is less than $\epsilon/2$. Let $U_i$ be an open interval of length less than $\epsilon/4n_m$ containing the left endpoint of $I_{n_i}$, and let $V_i$ be an open interval of length less than $\epsilon/4n_m$ containing the right endpoint of $I_{n_i}$. Let $$U_n=\bigcup_{i=1}^{n_m} U_i\cup V_i.$$ Then $U_n$ in an open set of measure less than $\epsilon/2$ which contains all of the endpoints of the intervals $I_{n_i}$. Finally, $A=(\text{int } J_n)\cup U_n$ is an open set containing the cantor set which has measure less than $\epsilon.$