Let $X$ be a locally compact complete separable metric space and $T:X \rightarrow X$ be continuous. Given $x \in X$, define $$\varphi_x(f) := \lim \frac{1}{n} \sum_{k = 0}^{n - 1} f(T^k(x))$$where $f \in C_b(X)$(only consider real codomains).
Assume that $\varphi_x(f)$ exists for every $f \in C_b(X)$. I have shown that $\varphi_x:C_b(X) \rightarrow \mathbb{R}$ is unital linear functional. I am confused about why we can use we can use Riesz Representation Theorem to assert that there exists a unique (Radon) probability measure $\mu$ such that $\varphi_x(f) = \int f d\mu$. Is this only for $C_c(X)$, does this work for $C_b(X)$? It wasn't that clear since in the notes that I am reading, he just stated that $\varphi_x(f) = \int f d\mu$. I understand how he is applying RRT.