Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $A_n\varphi(x)$ of Birkhoff averages converges as $n\to\infty$, where $$A_n\varphi(x)=\frac1n\sum_{j=0}^{n-1} \varphi(T^j(x)).$$
Setting $$L_x(\varphi)=\lim_{n\to\infty}A_n\varphi(x)$$ we obtain a bounded linear functional on the space $\mathcal{C}_b(X)$ of bounded continous functions $\varphi\colon X\to\mathbb{R}$. Clearly, $L_x(\varphi)=L_x(\varphi\circ T)$, but does $L_x$ have to be given by $$L_x(\varphi)=\int_X\varphi\,\text{d}\mu$$ for some $T$-invariant Borel probability measure $\mu$ on $X$?
Comments
Clearly, the answer is yes if we assume that $X$ is compact (by the Riesz theorem). Furthermore, if $\hat{X}$ is any compactification of $X$, the $L_x$ must be given by some Borel probability measure $\hat{\mu}$ on $\hat{X}$, but since it may be impossible to extend $T$ to $\hat{X}$ we cannot claim that $\hat{\mu}$ is $T$-invariant.