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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?

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?

Rudin's Real and Complex Analysis proves the theorem for the following two cases:

• For linear functionals on the space $C_K(X)$, 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?

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?

Update: Actually, in chapter six, he also treats the case of $C_0(X)$, the space of continuous functions on $X$ vanishing at infinity. $X$ is still locally compact and Hausdorff (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 $\mathbb R^k$.

Rudin's Real and Complex Analysis proves the theorem for the following two cases:

• For linear functionals on the space $C_K(X)$, 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?