THE STRUCTURE ON THE REAL FIELDGENERATED BY THE STANDARD PART MAP ON ANO-MINIMAL EXPANSION OF A REAL CLOSED FIELD

摘要

Let R be a sufficiently saturated o-minimal expansion of a real closedfield,let Obe the convex hull of Q in R,and let st:O~n→R~n be thestandard part map.ForXR~ndefine stX:=st(X O~n).We letR_(ind)be the structure with underlying set R and expanded by all sets st X,where XR~nis definable inRandn=1,2,.... We show that thesubsets of R~nthat are definable in R_(ind)are exactly the finite unions ofsets stXst Y,where X,Y R~nare definable inR.A consequence ofthe proof is a partial answer to a question by Hrushovski,Peterzil andPillay about the existence of measures with certain invariance propertieson the lattice of bounded definable sets in R~n.