Automorphism groups of prime models, and invariant measures

We adapt the notion from [7] and [2] of a (relatively) definable subset of A u t ( M ) when M is a saturated structure, to the case A u t ( M / A ) when M is atomic and strongly ω -homogeneous (over a set A ). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of A u t ( M / A ) . For example when T is stable, we have existence and uniqueness. We also discuss the compatibility of our definability notions with definable Galois cohomology from [12] and differential Galois theory.