Let $\alpha\notin\mathbb{Q}$ and define $T:[0,1]\to[0,1]$ by $T(x)=x+\alpha\,{\rm mod}\,1$. I already know that T is ergodic. I need to show that for any $f\in C([0,1])$, we get the pointwise limit: $$ \lim\limits_{N\to\infty}\frac{1}{N+1}\sum\limits_{n=0}^{\infty}f(T^n(x))=\int\limits_0^1f(t){\rm d}t $$ I tried using approximations by first showing it to trigonometric polynomials. Namely I showed it works for $f(x)=e^{ikx}$, and then by linearity for trigonometric polynomials. I thought I could use Stone-Weirstrass to now proove this fact for continous periodic functions on $[0,1]$, but I get stuck, and I am honestly quite unsure about why should it be true in the first place. Since for every trigonometric polynomial I get $0$ in the limit, why should I expect anything else for other continous functions?
And obviously there are continous functions for which this limits is not $0$, so I'm pertty sure it's a bad sign for me.
If possible, I am looking for another angle to tackle this question.