On first order amenability

Abstract We introduce the notion of first order amenability , as a property of a first order theory T : every complete type over $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> , 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 $${{\,\mathrm{{Aut}}\,}}(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mi>Aut</mml:mi> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for all sufficiently large $$\aleph _{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -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 $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> coincide. This extends and essentially generalizes a similar result proved via different methods for $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -categorical theories in Adv. Math. 345 , 1253–1299 (2019). In the special case when amenability is witnessed by $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable global Keisler measures (which is for example the case for amenable $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -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 .