Operations that preserve integrability, and truncated Riesz spaces

TL;DR

This paper characterizes operations preserving p-integrability in measure spaces, linking them to Dedekind σ-complete truncated Riesz spaces, and constructs concrete models of free such spaces.

Contribution

It provides a complete characterization of operations that preserve p-integrability and establishes their connection to Dedekind σ-complete truncated Riesz spaces, including explicit models.

Findings

Operations preserving p-integrability are exactly those in the Dedekind σ-complete truncated Riesz spaces.

The variety of these spaces is generated by the real numbers, nd concrete free models are constructed.

Analogous results hold for finite measure spaces with Dedekind omplete Riesz spaces with weak unit.

Abstract

For any real number $p\in [1,+\infty)$, we characterise the operations $\mathbb{R}^I \to \mathbb{R}$ that preserve $p$-integrability, i.e., the operations under which, for every measure $\mu$, the set $\mathcal{L}^p(\mu)$ is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind $\sigma$-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that $\mathbb{R}$ generates this variety. From this, we exhibit a concrete model of the free Dedekind $\sigma$-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve $p$-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind $\sigma$-complete Riesz spaces with weak unit, $\mathbb{R}$ is proved to generate this variety, and a concrete model of the free Dedekind $\sigma$-complete Riesz spaces with weak unit is exhibited.