Rotation on circle ergodic measures

Question

Let $T:\mathbb{S}^1\to\mathbb{S}^1$, $T(x):=x+\alpha\;\;\text{(mod}\,1)$ with $\alpha\in\mathbb{Q}$. Then we know that every $x\in \mathbb{S}^1$ is periodic with period $q$.

1. Show that the measures $$m_x=\frac{1}{q}(\delta_x+\delta_{T(x)}+\dots +\delta_{T^{q-1}(x)})$$ are ergodic.
2. Show that these are in fact the only ergodic probability measures.