Is there a nice theory of Keisler measures in simple theories, or is it just the abstract measure theory?  Although very well behaved measures in simple theories arise naturally in examples (e.g. in pseudo-finite structures), little is known in general. There are a few positive results (Keisler randomization, higher arity generalization of the VC theory in the n-dependent case), and a few negative ones. In this talk we focus on translation invariant measures in definable groups. I will discuss the theory of definable paradoxical decompositions, and describe a method of expanding $SL_2(C)$ (or any definable group containing a non-abelian free subgroup) by a generic paradoxical decomposition that preserves simplicity of the theory. This answers a well-known question in the area: there are groups in simple theories that are not definably amenable. We also discuss some applications in small theories, including nontriviality of their graded Grothendieck rings.

Joint work with Hrushovski, Kruckman, Krupinski, Moconja, Pillay and Ramsey.