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Let R be a su-ciently saturated o-minimal expansion of a real closed fleld, let O be the convex hull of Q in R, and let st : On !Rn be the standard part map. For X µ Rn deflne stX := st(X \ On). We letRind be the structure with underlying setR and expanded by all sets of the form stX, where X µ Rn is deflnable in R and n = 1;2;:::. We show that the subsets of Rn that are deflnable in Rind are exactly the flnite unions of sets of the form stX n stY , where X;Y µ Rn are deflnable 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 deflnable sets in Rn.

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