# Open dense subset with small outer measure

Show that for any $\delta>0$ there exists an open dense subset $U$ of $\mathbb{R}$ with $\mu(U)<\delta$, where the measure is the outer measure.

I'm not sure how to go here. The set $U$ must be dense (so for any point $x\in\mathbb{R}$ there exists a point in $U$ arbitrarily close to $x$), open (so any point in $U$ has a neighborhood entirely contained in $U$), and yet can be covered by a countable union of intervals with really small length.

$\mathbb{Q}$ is dense. Let $\{q_n : n \in \mathbb{N}\}$ enumerate $\mathbb{Q}$. Let $U = \bigcup_{n \in \mathbb{N}} B(q_n,\frac{\delta}{2^n})$. $U$ is open and has measure less than $\delta$. It is dense because $\mathbb{Q} \subseteq U$.