As mentioned in the title, the question is how to show (using Vitali Covering Lemma) that the union of any collection of closed, bounded non-generate intervals is measurable.
The Vitali Covering Lemma as stated in my notes is: Let $E$ be a subset of $\mathbb{R}$ with $m^*(E)<\infty$ and let $\mathcal{V}$ be a Vitali covering of $E$. Then, for each $\epsilon>0$, there is a finite disjoint sequence $(J_k)_{k=1}^n$ of intervals in $\mathcal{V}$ such that $m^*(E\setminus \cup_{k=1}^n J_k)<\epsilon$.
I am stuck with what to choose as "$E$". I was about to choose $E$ as the union of the collection of intervals, but realised that $m^*(E)$ may be infinite, and hence the conditions for using the lemma are not met. I also thought about choosing $E$ as a single interval, but Vitali doesn't seem to help here since an uncountable union of finite intervals is still uncountable. (Initially, I was worried that Vitali covering may not exist, but I managed to figure that a Vitali covering always exists, just take the set of all closed, bounded, nondegenerate intervals).
Thanks for any help.