Appears in collection : Model Theory and Valued Fields

In an instance of motivic integration, Hrushovski and Kazhdan study the definable sets in the theory of algebraically closed valued fields of characteristic zero. They show that the Grothendieck group K(VF) of definable sets in the sort VF is isomorphic to the one of the RV sort modded out by an explicit relation. I will show how this construction relates to an equivalence by Ayoub between the category of motives of rigid analytic K-varieties, for K = k((t)), and the category of quasi-unipotents motives over k. In particular I will build a ring morphism from K(VF) to the Grothendieck ring of rigid analytic motives.