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?