A measure for perfect PAC fields with pro-cyclic Galois group

In “Pseudo-finite fields and related structures” Hrushovski asks (in a more general context) whether the notion of measure on definable sets in pseudo-finite fields can be extended to perfect PAC fields whose Galois group is bounded but not . We define a suitable generalization of measure and give an answer to this question: yes if the Galois group is pro-cyclic, no otherwise.