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) Aut ( M ) for all sufficiently large ₀ 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 theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from Selecta Math. (N.S.) 28, 16 (2022), 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 ω -categorical theories in Adv. Math. 345 , 1253–1299 (2019). In the special case when amenability is witnessed by ∅ ∅ -definable global Keisler measures (which is for example the case for amenable ω 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 .