Measure preserving transformations arise in many different settings. Each
setting gives its own topology on the collection of transformations and
some provide algebraic structure as well.

A natural question is whether two different settings have the same generic
dynamical properties and give the same Borel structure on the measure
preserving transformations.

Dan Rudolph gave a meta-conjecture that all settings are equivalent. In
these two talks we make this precise in various ways and prove it. We also
introduce some new settings such as the space of rational invariant
measures.