I have a set $X=\{1,2,...,N\}$ and the map $T:X \to X$: $Ti=i+1 \text{ mod } N$. Now I want to show that $T$ is uniquely ergodic and find the unique measure.
I know it holds that $T^Nx=x$ iff $\frac{1}{N} \sum^{N-1}_{i=0} \delta_{T^ix} \in M(X,T)$. So I know for sure that $\upsilon=\frac{1}{N} \sum^{N-1}_{i=0} \delta_{T^ix}$ is a $T$ invariant probability measure.
So now I want to show that for all arbitrary $T$ invariant probability measure $m$ holds that $m=\upsilon$.
Does someone know how to do this?
If $m$ is an invariant measure, then $m(x) = m(T^{-1}(x)) = m(x-1)$ for all $x$, where $x-1$ is understood modulo $N$, and $m(x) = m(\{x\})$. This implies that $m(x) = m(y)$ for all $x,y$.