Let $d\geq1$ and let $\mu$ be a regular Borel measure on $\mathbb{R}^d$. Show that $L^2(\mu)$ is separable.
I seem to have reduced the problem to showing that indicator functions of cubes with rational coordinates can be made arbitrarily close to indicator functions of bounded measurable sets, but I am unable to proceed further.
Let $D$ be the collection of finites unions of cubes with rational coordinates. This collection is countable, and we will show that it is dense.
Due to the regularity of $\mu$, it suffices to approximate the indicator function of an open set. An open set is a countable union of open cubes with rational coordinates. Let $O$ be an open set of finite measure and let $\left(R_n\right)_{n\in\mathbb N}$ be a sequence of open cubes with rational coordinates such that $O=\bigcup_{n\in\mathbb N}R_n$. Let $O_N:=\bigcup_{n=1}^NR_n$. Then $O_N$ belongs to $D$ and the indicator of $O_N$ converges almost everywhere to that of $O$. Conclude by dominated convergence.