Invariant measures in simple and in small theories

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over ∅ but has 𝜇-measure 0 for every automorphism invariant Keisler measure 𝜇 and (ii) a definable group 𝐺 in a simple theory such that 𝐺 is not definably amenable, i.e. there is no translation invariant Keisler measure on 𝐺. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups.