Beta transformation is Ergodic. Let $\beta \in \mathbb{R}$ with $\beta >1$. Define $T_{\beta}:[0,1)\to [0,1)$ by:
$$T_{\beta}(x)=\beta x-[\beta x]=\{\beta x\} $$.
Consider:
$$ h(x)=\sum_{n=0}^{\infty}{\chi_{\{y:y<T_{\beta}^n(1)\}}(x)}$$
Show that the map $T_{\beta}$ preserves the measure $\mu$ defined by:
$$\mu(A)=\int_{A}{h(x)dx}   $$
And then prove that $T_{\beta}$ is ergodic with respect to $\mu$.
I have some question regarding to this problem. First of all, I want to know some properties of the orbit $\{T_{\beta}^n(1), n\ge 0 \}$. I think that this measure is finite but I could not prove it.
Unfortunately, I could not prove anything directly related to the problem =/

Are you sure that $h$ looks exactly like this? I would expect $\chi$ to have coefficients like $\beta^{-n}$ which would help convergence. Also if $1$ has finite orbit (is periodic) you have to take a finite sum. 
Proof of the invariance is a straightforward check using Perron-Frobenius operator. Again using PF operator you can easily check ergodicity criteria (for example you can show that Lebesgue measure (density is constant equal to $1$) converges to $h$ under its action).
