We let R be an o-minimal expansion of a field, V a convex subring, and (R 0 , V 0 ) an elementary substructure of (R, V ). Our main result is that (R, V ) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R, provided that k R (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k with structure induced from (R, V ). We also give a criterion for a superstructure of (R, V ) being an elementary extension of (R, V ).

Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st : V → k. Let k ind be the expansion of k by the standard parts of the definable relations in 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 ω-saturated and V is the convex hull of Q in R, then the sets definable in k ind are exactly the standard parts of the sets definable in (R, V ).

We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field R. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of R. We show that every measurable subset of R n with non-empty interior has positive measure, and that the measure is preserved by definable C 1 -diffeomorphisms with Jacobian determinant equal to ±1.

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of Q in R, and let st : O n → R n be the standard part map. For X ⊆ R n define st X := st (X ∩ 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 R n 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 .

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T -convex theory.