The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

Abstract

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We let \mathbb{R}_{\ind} be the structure with underlying set \mathbb{R} and expanded by all sets of the form st(X), where X \subseteq R^{n} is definable in R and n=1,2,.... We show that the subsets of \mathbb{R}^n that are definable in \mathbb{R}_{\ind} are exactly the finite unions of sets of the form st(X) \setminus st(Y), where X,Y \subseteq R^n are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in R^n.