Measuring definable sets in o-minimal fields

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.