Definable amenability is the property of a definable group 
for which there is a Keisler measure 
on the definable sets contained
in 
that is invariant under left translation, so
 |
for every 
and every definable set 
. Thus definable amenability is the model
theory analogue of the existence of an invariant probability
measure.
For groups in theories without the independence property, definable amenability is equivalent to the existence of a global type with bounded left-translation orbit (Chernikov and Simon 2018). This equivalence fails for arbitrary theories, as shown by the counterexample to Petrykowski's conjecture.
