Abstract: This is work in progress, joint with Nick Ramsey (UCLA).
A conjecture, now disproved by Chernikov, Hrushovski, Kruckman, Krupinski, Pillay and Ramsey, asked whether any group with a simple theory is definably amenable. It is well known that the counting measure on finite fields gives rise to a non-standard counting measure on pseudo-finite fields (the infinite models of the theory of finite fields). It was unknown whether other PAC fields possessed a reasonable measure, and in this talk, we will show that some of them do, although the measure we define does not have all the nice properties of a counting measure when the field is not pseudo-finite. This result can be used to show that if G is a group definable in an e-free perfect PAC field, then G is definably amenable.
We hope to extend our results to the wider class of bounded perfect PAC fields.