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.