It is well known that when X is compact space (or locally compact space), $C(X)=\{f:f X\rightarrow X \text{is continous,and bounded} \}$ ,the dual space of $C(X) C(X)^{*} $is correspond to $M(X)$ space of Radon measure with bounded variation.

However according to my knowledge. there are few books discuss case that when X is noncompact, for example complete vsperable metric space.

even for the simplest example, when taking X is R, means real line, what does $(C(X))^{*}$ mean.

Any advice and reference will be appreaciated.