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.