Nonlinear Lebesgue spaces: Dense subspaces, completeness and separability

TL;DR

This paper systematically studies nonlinear Lebesgue spaces, establishing their measure-theoretic properties, including completeness, separability, and density of subspaces, extending classical linear results to nonlinear settings.

Contribution

It provides the first comprehensive analysis of measure-theoretic properties of nonlinear Lebesgue spaces, unifying and extending existing results from the linear case.

Findings

Characterization of completeness and separability of nonlinear Lebesgue spaces.

Proof of density of simple, continuous, and smooth mappings in these spaces.

Extension of classical linear space results to nonlinear metric-valued function spaces.

Abstract

L^p spaces of mappings taking values in arbitrary metric spaces, which we call nonlinear Lebesgue spaces, play an important role in several fields of mathematics. For instance, membership in these spaces is typically required for transport maps in optimal transport theory and for stochastic processes in probability theory. Nonlinear Lebesgue spaces also arise naturally in applications such as medical imaging, where the physical signals at play often exhibit little regularity and take their values in nonlinear spaces. Yet, these spaces remain little studied in the literature, likely due to their lack of differential structure outside the case where mappings are valued in a linear space. This paper is the first in a series by the authors devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces. The present article exposes a systematic treatment of their measure-theoretic properties, unifying and refining scattered results from the literature while also extending classical results from the linear setting to this broader nonlinear framework -- including the characterizations of their completeness and their separability as well as the density of some of their subspaces: the spaces of simple, continuous and smooth mappings.