$X$ is locally compact hausdorff space. Let $\Gamma$ denote the set of positive linear functional on $C_c(X)$(i.e. all continuous functions over $X$ with compact support.) Denote $M(X)$ set of Borel measure over $X$ where measures might be extendable to larger set.
From Riesz representation theorem, I deduce $\Gamma\to M(X)$ as $R_{>0}$ linear map is injection map as every positive linear functional pins down a unique Borel measure. Clearly I can define $M(X)\to\Gamma$ by $\mu\to \int(-)d\mu$.
$\textbf{Q:}$ Do I have $\Gamma\to M(X)\to\Gamma$ and $M(X)\to\Gamma\to M(X)$ both identity maps in general? As far as I could see on Rudin's proof(Thm 2.17,2.18 3rd Ed Real and Complex Analysis), I need $X$ $\sigma$-compact to have inner regularity which pins down identity map $M(X)\to\Gamma\to M(X)$.