Let $(X,\mathcal{M},\mu)$ be a measure space. Take $\epsilon > 0$ and $A \in \mathcal{M}$.
Is it in general true that there exist and open set $U$ such that $A \subseteq U$ and $\mu(U \setminus A)<\epsilon$ ?
If it is, I would also be grateful for a hint on how to prove this.
I need only the case when $X=\mathbb{R^n}$, $\mathcal{M} = \mathcal{B}(\mathbb{R^n})$, and $\mu$ is the Lebesgue measure. But I would be interested in a more general result, if there is one.
Thank you in advance!