On a Lebesgue-like integral over the Levi-Civita field $\mathcal{R}$

TL;DR

This paper introduces a new Lebesgue-like integral over the Levi-Civita field, extending integration theory to a non-Archimedean setting while preserving key properties of classical integrals.

Contribution

It develops a novel integration framework over the Levi-Civita field that generalizes previous work and overcomes the challenge of non-existence of infima and suprema for bounded sets.

Findings

The family of measurable functions forms an algebra closed under absolute values.

The integral is linear, countably additive, and monotone.

The integral of a non-negative function is zero iff the function is zero almost everywhere.

Abstract

The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In this paper we develop a new theory of integration over $\mathcal{R}$ that generalizes previous work done in the subject while circumventing the fact that not every bounded subset of $\mathcal{R}$ admits either an infimum or a supremum. We define a new family of measurable functions and a new integral over measurable subsets of $\mathcal{R}$ that satisfies some very important results analogous to those of the Lebesgue and Riemann integrals for real-valued functions. In particular, we show that the family of measurable functions forms an algebra that is closed under taking absolute values, that the integral is linear, countably additive and monotone, that the integral of a non-negative function $f:A\to\mathcal{R}$ is zero if and only if $f=0$ almost everywhere in $A$ and that the analogous versions for the uniform convergence theorem and the fundamental theorem of calculus hold true.