Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem

We study relatively invariant measures with the multiplicators ${\mathbb Q}^*_+\ni q\mapsto q^{-\beta}$ on the space $\mathcal A_f$ of finite adeles. We prove that for $\beta\in(0,1]$ such measures are ergodic, and then deduce from this the uniqueness of KMS$_\beta$-states for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of $\mathbb Q^*$ on the full adeles.