  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 $\pm 1$.
