Actions of Amenable Equivalence Relations on CAT(0) Fields

After a short introduction to the general notion of Borel fields of metric spaces we introduce the notion of the action of an equivalence relation on such fields. Then, we specify the study to the Borel fields of proper CAT(0) spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.