Lebesgue measure theory and integration theory on non-archimedean real closed fields with archimedean value group

Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the measure and integration is obtained in a controled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field.