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We study relatively invariant measures with the multiplicators Q* + q? q -β on the space A f of finite adeles. We prove that for β∈ (0, 1] such measures are ergodic, and then deduce from this the uniqueness of KMS β -states for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of Q* on the full adeles.

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