Groups of measurable functions

Let G be a Polish group, and let m be a non-atomic probability measure on the interval  [0,1].  We define L_0(G) to be the Polish group of all m-equivalence classes of m -measurable functions with values in G. The multiplication in L_0(G) is pointwise, and the the topology is of convergence in measure.
In this talk, I will be interested in three notions (either on the level of G, or L_0(G)): ample generics, topological similarity, and cyclically dense conjugacy classes. This is joint work with Aleksandra Kwiatkowska.