Spaces of measurable functions on the the Levi-Civita field

TL;DR

This paper develops and analyzes spaces of measurable functions over the Levi-Civita field, introducing new Banach and Hilbert space structures, and explores their applications to distributions and generalized functions.

Contribution

It defines $ ext{L}^p$ spaces over the Levi-Civita field, studies their completions, and establishes a duality framework for measurable functions representing distributions.

Findings

The $ ext{L}_s^p$ spaces are Banach spaces.

The $ ext{L}_s^2$ space can be endowed with an inner product, forming a Hilbert space.

The framework effectively models distributions like Dirac and Heaviside.

Abstract

We introduce the $\mathcal{L}^p$ spaces of measurable functions whose $p$-th power is summable with respect to the uniform measure over the Levi-Civita field $\mathcal{R}$. These spaces are the counterparts of the real $L^p$ spaces based upon the Lebesgue measure. Nevertheless, they lack some properties of the $L^p$ spaces: for instance, the $\mathcal{L}^p$ spaces are not complete with respect to the $p$-norm. This motivates the study of the completions of the $\mathcal{L}^p$ spaces with respect to strong convergence, denoted by $\mathcal{L}_s^p$. It turns out that the $\mathcal{L}_s^p$ spaces are Banach spaces and that it is possible to define an inner product over $\mathcal{L}_s^2$, thus making it a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of real measurable functions as sequences of measurable functions in $\mathcal{R}$ that weakly converge in measure. We also define a duality between measurable functions and representatives of continuous functions. This duality enables the study of some measurable functions that represent real distributions. We focus our discussion on the representatives of the Dirac distribution and on the well-known problem of the product between the Dirac and the Heaviside distribution, and we show that the solution obtained with measurable functions over $\mathcal{R}$ is consistent with the result obtained with other nonlinear generalized functions.