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.



Why is this generality in Vitali's Lemma useful?
The short answer (to the question in the title) is so that you don't need to worry about whether E is measurable. If you happen to know that E is measurable then you can drop "outer" everywhere.


There may be a longer answer (better addressing the question as stated in the page) involving specific applications where E is in fact nonmeasurable, but I personally don't know of such applications offhand.


