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

TL;DR

This paper characterizes the structure generated on the real field by the standard part map in o-minimal expansions of real closed fields, revealing its definable sets as finite unions of specific standard part images.

Contribution

It provides a precise description of the definable sets in the structure expanded by standard parts, answering a question about measures on bounded definable sets.

Findings

Definable sets in the expanded structure are finite unions of standard part images minus other such images.

The structure captures exactly the sets of the form st(X) extbackslash st(Y) for definable X,Y.

Partial progress on a question about invariant measures on bounded definable sets.

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.