**Update:** Actually, in chapter six, he also treats the case of $C_0(X)$, the space of continuous functions vanishing at infinity (Theorem 6.19). Since this is the most general form of the theorem I know, surely this will suit your purposes?

In *Real and Complex Analysis*, as far as I know, Rudin only proves the Riesz Representation theorem for linear functionals on the space $C_K(X)$, where $X$ is a locally compact Hausdorff space.

Offhand comment:
For the purposes of the particular chapter (The second one) and even the book, this is akin to nuking mosquitoes, since you need much less to construct the Lebesgue measure on $\mathbb R^k$.