I've spent a few hours trying to show that $m\left(E-\cup_{n=1}^N I_n \right)<\epsilon$ where the $I_n$'s are a bunch of disjoint closed intervals with $I_n\subset E$ and $E\in \mathbb{R}: E$ has finite measure.
The farthest I can get is showing that $m(E)-m(\cup_{n=1}^N I_n)$. Instinctively, I know that $\cup_{n=1}^N I_n \leq \sum^{\infty}_{n=1} m(I_n)$, but, this is not a finite sum. At this point I'm not even sure that this is a true statement. Any thoughts?
Standard counterexample: $E$ is the set of irrational numbers in $[0,1]$. $m(E)=1$ but there are no nontrivial closed intervals that are subsets of $E$.