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

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.