Let $\mathbb{R}^n$ have the Borel sigma algebra given by the standard topology
Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be uniquely ergodic
where $\mu$ is the unique ergodic measure
Assume $\mu$ has bounded support and let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a continuous function
Must the following hold for all $x\in\mathbb{R}^n$?
$\lim_{T\rightarrow\infty}\frac{1}{T}\sum_{k=0}^Tf(T^kx) = \int fd\mu$
Note that this is not a direct result Birkhoffs Theorem because I am interested in all $x\in\mathbb{R}^n$.