On first order amenability
We introduce the notion of first order amenability, as a property of a first order theory T: every complete type over ∅, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of T follows from amenability of the (topological) group Aut(M) for all sufficiently large ℵ_0-homogeneous countable models M of T (assuming T to be countable), but is radically less restrictive. First, we study basic properties of amenable
... ries, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if T is amenable, then T is G-compact, namely Lascar strong types and Kim-Pillay strong types over ∅ coincide. This extends and essentially generalizes a similar result proved via different methods for ω-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by ∅-definable global Keisler measures (which is for example the case for amenable ω-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.