TITLE: Is $M(X)$ (regular borel measures), the dual of $C_0(X)$ separable?
QUESTION [5 upvotes]: For $X$ locally compact (let's take $X=\mathbb{R}^d$), we know that the dual of $C_0(X)$ is $M(X)$, the space of regular borel measures on X. $C_0(X)$ is separable but is $M(X)$ separable? I have tried searching but haven't seen the result nowhere, so I think not.

REPLY [7 votes]: As long as $X$ is countable, $M(X)$ is separable. If $X$ is countable, the dual is isomorphic to $\ell_1$ hence separable. If $X$ is uncountable, Dirac measures form an uncountable, discrete set. It cannot happen in a separable metric space.