| Résume | We describe the background and outline a proof of the following theorem: for a generic measure preserving transformation $T$, the closed group generated by $T$ is not isomorphic to the topological group $L^0(\lambda, {\mathbb T})$ of all Lebesgue measurable functions from $[0,1]$ to $\mathbb T$. This result answers a question of Glasner and Weiss. The main step in the proof consists of showing that Koopman representations of ergodic boolean actions of $L^0(\lambda, {\mathbb T})$ possess a non-trivial property not shared by all unitary representations of $L^0(\lambda, {\mathbb T})$. In proving that theorem, an important role is played by a new mean ergodic theorem for ergodic boolean actions of $L^0(\lambda, {\mathbb T})$. |
