Rudin's Real and Complex Analysis proves the theorem for the following two cases where $X$ is a locally compact Hausdorff space. :
For linear functionals on the space $C_c(X)$, the space of all continuous compactly supported functions. (Theorem 2.14)
For linear functionals on the space $C_0(X)$, the space of all continuous functions vanishing at infinity. (Theorem 6.19)
Since the second is the most general form of the theorem I know, surely this will suit your purposes?
