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

TL;DR

This paper develops a Lebesgue measure and integration theory for semialgebraic sets over non-archimedean real closed fields with archimedean value groups, extending classical results using model theory and valuation techniques.

Contribution

It introduces a full measure and integration framework in a non-archimedean setting, generalizing classical Lebesgue theory to new algebraic structures.

Findings

Measure and integration theory established for semialgebraic sets in non-archimedean fields.

Construction determined by valuation sections, with uniqueness in the rational value group case.

Range of measure and integration is controlled and related to the original real closed field.

Abstract

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.