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 $\mathcal{O}$ be the convex hull of ℚ in R, and let st: $\mathcal{O}^n$ → ℝⁿ be the standard part map. For X ⊆ R ⁿ define st X:= st (X ∩ $\mathcal{O}^n$). We let ℝ_ind be the structure with underlying set ℝ and expanded by all sets st X, where X ⊆ R ⁿ is definable in R and n = 1, 2,.... We show that the subsets of ℝⁿ that are definable in ℝ_ind are exactly the finite unions of sets st X st Y, where X, Y ⊆ 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.