It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f: X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of Radon measures with bounded variation.

However, according to my knowledge, there are few books which discuss the case when X is noncompact, for example a complete separable metric space.

Even for the simplest example, when taking $X=\mathbb{R}$, what does $(C(X))^{*}$ mean?

Any advice and reference will be much appreciated.