I have a bounded set $E$ of real numbers. I'm in the process of showing that there is a set $G$ that is a countable intersection of open sets $G_i$ such that $E\subseteq G$ and $E,G$ have the same outer measure.
What I have so far: I know that outer measure is invariant under unions of disjoint sets, so I am characterizing $E$ as a union of open intervals, closed intervals, half-open intervals, and unions of countably many singletons as well as unions of collections of uncountably many singletons.
Open intervals of $E$ can be replaced by $G_i$ for some $i$. Closed intervals of $E$, say $[a,b]$ can be written as $\bigcap _{i=1}^\infty (a-\frac{1}{i},b+\frac{1}{i})$, half-open intervals $[a, b)$ can be written as $\bigcap _{i=1}^\infty(a-\frac{1}{i},b)$ and a similar statement can be made for half-open intervals of the form (a,b].
I'm stuck on the singletons since I have to make sure I use only countably many sets. Any suggestions? I know that the outer measure of a collection of uncountably many points is equal the measure of the smallest open interval containing the points, but I'm having the hardest time given a point determining whether I want it to belong to a countable or uncountable set.