Let S be a lebesgue measurable subset of R and let lambda denote the lebesgue measure. Show that there exists sets $H$ and $K$ such that
1) $S \cup H$ is open and $\lambda(H) < \epsilon$.
2) $S \backslash K$ is clsoed and $\lambda(K) < \epsilon$
for 1): I have set $H = \cup_i^\infty(a_i,b_i) \backslash S$ and we have for any set $S$ $\lambda(\cup_i^\infty(a_i,b_i)) < \lambda(S) + \epsilon$. This was enough to get the result only for $\lambda(S) < \infty$. I am not sure what to do if $\lambda(S) = \infty$ or for 2)