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 $\pm 1$. 
Jana Ma\v{r}\'ikov\'a and Masahiro Shiota
math.LO