  Let $\pi$ be an automorphism of the Turing degrees induces by a homeomorphism
$\varphi$ of the Cantor space $2^\omega$ such that $\varphi$ preserves all
Bernoulli measures. It is proved that $\pi$ must be trivial. In particular, a
permutation of $\omega$ can only induce the trivial automorphism of the Turing
degrees.
