On a new measure on the Levi-Civita field $\mathcal{R}$

TL;DR

This paper introduces a new measure on the Levi-Civita field that improves upon previous definitions, ensuring better alignment with classical measure theory and expanding the class of measurable sets.

Contribution

It characterizes measurable sets in the Levi-Civita field and develops an outer measure leading to a more comprehensive measure theory.

Findings

The new measure generalizes Lebesgue measure to the Levi-Civita field.

Most classical properties of Lebesgue measure hold under the new measure.

The family of measurable sets is strictly larger than previous definitions.

Abstract

The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [Shamseddine-Berz-2003], a measure was defined on $\mathcal{R}$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over $\mathcal{R}$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [Shamseddine-Berz-2003]. Then we will introduce the notion of an outer measure on $\mathcal{R}$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on $\mathcal{R}$ that proves to be a better generalization of the Lebesgue measure from $\mathbb{R}$ to $\mathcal{R}$ and that leads to a family of measurable sets in $\mathcal{R}$ that strictly contains the family of measurable sets from [Shamseddine-Berz-2003], and for which most of the classic results for Lebesgue measurable sets in $\mathbb{R}$ hold.