A real-valued measure on non-Archimedean field extensions of $\mathbb{R}$

TL;DR

This paper introduces a new real-valued measure on non-Archimedean fields extending the reals, enabling canonical measurable representatives for Lebesgue sets and comparing with existing measures, with applications to integration and distributions.

Contribution

It defines a novel real-valued measure on non-Archimedean fields, generalizing Lebesgue measure, and establishes its properties and relationships with existing measures, including the Levi-Civita field.

Findings

The measure assigns equal measure to Lebesgue measurable sets and their non-Archimedean representatives.

The introduced measure is infinitesimally close to the Shamseddine-Berz measure on the Levi-Civita field.

Every continuous function on the Levi-Civita field has an integrable representative under the new measure.

Abstract

We introduce a real-valued measure ${m_L}$ on non-Archimedean ordered fields $(\mathbb{F},<)$ that extend the field of real numbers $(\mathbb{R},<)$. The definition of ${m_L}$ is inspired by the Loeb measures of hyperreal fields in the framework of Robinson's analysis with infinitesimals. The real-valued measure ${m_L}$ turns out to be general enough to obtain a canonical measurable representative in $\mathbb{F}$ for every Lebesgue measurable subset of $\mathbb{R}$, moreover, the measure of the two sets is equal. In addition, $m_L$ it is more expressive than a class of non-Archimedean uniform measures. We focus on the properties of the real-valued measure in the case where $\mathbb{F}=\mathcal{R}$, the Levi-Civita field. In particular, we compare ${m_L}$ with the uniform non-Archimedean measure over $\mathcal{R}$ developed by Shamseddine and Berz, and we prove that the first is infinitesimally close to the second, whenever the latter is defined. We also define a real-valued integral for functions on the Levi-Civita field, and we prove that every real continuous function has an integrable representative in $\mathcal{R}$. Recall that this result is false for the current non-Archimedean integration over $\mathcal{R}$. The paper concludes with a discussion on the representation of the Dirac distribution by pointwise functions on non-Archimedean domains.