An additive measure in o-minimal expansions of fields

Given an o-minimal structure M which expands a field, we define, for each positive integer d, a real-valued additive measure on a Boolean algebra of subsets of M^d and we prove that all the definable sets included in the finite part Fin(M^d) of M^d are measurable. When the domain of M is the real line we obtain Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets (the Jordan measurable sets). Our measure has good logical properties, being invariant under elementary extensions and under expansions of the language. In the final part of the paper we consider the problem of defining an analogue of the Haar measure for definably compact groups.