Hausdorff measure on o-minimal structures

In [Berarducci-Otero, 2004], the authors define an analogue of Lebesgue measure for bounded definable subsets of an o-minimal structure K , which expands a real closed field. Our aim is defining the area of a bounded definable d-dimensional subsets of Kn. Different possible definitions for such area are possible: we will prove that some of them are equivalent. On the reals, there are many different definitions of the d-area of a subset. However, on smooth sub-manifolds of Rn all these definitions coincide with the d-dimensional Hausdorff measure Hd . Every definable set in an o-minimal structure on the reals is a finite union of smooth sub-manifolds, therefore the d-area of such a set is Hd . Many formulae are known for the Hausdorff measure: from Fubini’s theorem to the area and coarea formulae. We will generalise some of these formulae to our measure on K . Elisa Vasquez Rifo obtained independently some of the results exposed here.