Asger Törnquist: The lifting problem for the group of measure preserving transformations of the unit interval

Week 2, Thursday June 25, 16:30-17:20

A measure preserving transformation of the unit interval is an equivalence class of Borel measure preserving bijections that agree almost everywhere. The measure preserving transformations form a Polish group Aut([0,1],lambda), which may be considered as a dynamical analogue of the measure algebra. A measure preserving "near-action" of a group G on [0,1] is a homomorphism of G into Aut([0,1],lambda).
In this talk we will show that under CH, every near-action can be realized by a pointwise action by Borel measure preserving automorphisms on [0,1]. I will then discuss the possibility of having a model in which the near-action of Aut([0,1],lambda) on [0,1] itself does not have a point-wise realization, as well as a delimitative result in this direction due to Glasner, Weiss and Tsirelson.