In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?

**Vitali's Lemma:**
Let $E$ be a set of finite outer measure and $G$ a collection of intervals that cover $E$ in the sense of Vitali. Then given $\varepsilon> 0$ there is a finite disjoint collection of intervals in $G$ such that $m^*(E - \bigcup_{n=1}^N I_n) < \varepsilon$.

I'm trying to learn this theorem and I keep replacing "outer measure" with "measure" and I want a reason to stop doing that.