I'm having trouble starting two similar proofs:
Let $\epsilon > 0$. And let $E$ be a measurable set of finite measure.
Prove that there is an open set $U$ containing $E$ such that $m(U \setminus E) < \epsilon$.
Similarly, prove there is a compact set $K$ contained in $E$ such that $m(E \setminus K) < \epsilon$.
Any hints are much appreciated.
NOTE: $m$ is the Lebesgue outer measure. And $E \subseteq \mathbb{R}$ is measurable if $m(A) \geq m(A \cap E) + m(A \setminus E)$.