Abstract: Let R be an o-minimal field and V a proper convex subring of R with residue field k. Let k_{ind} be the expansion of the residue field by the standard parts of definable relations on R. We investigate the definable sets in k_{ind} and conditions on (R,V) which imply o-minimality of k_{ind}. We also show that if R is \omega-saturated and V is the convex hull of the rationals in R, then the sets definable in k_{ind} are exactly the standard parts of the sets definable in (R,V). Using our description of definable sets in k_{ind} we give a partial answer to a question posed by Hrushovski, Peterzil and Pillay about the existence of measures with a certain invariance property on the lattice of bounded definable sets in R^n.

Notes: A just-finished PhD student of van den Dries.