| Résume | Any measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action: an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of locally compact groups, while spatial actions of a certain class of large groups can only be trivial. |
