The sup-completion of a Dedekind complete vector lattice

TL;DR

This paper systematically studies the sup-completion of Dedekind complete Riesz spaces, extending classical results, introducing finite and infinite parts, and applying these to classical probability lemmas.

Contribution

It introduces a new framework for understanding the sup-completion of Dedekind complete Riesz spaces, including finite and infinite parts, and extends classical results like the Borel-Cantelli Lemma.

Findings

Established a systematic study of the sup-completion cone.

Extended classical results to the setting of Riesz spaces.

Presented applications including a Riesz space version of the Borel-Cantelli Lemma.

Abstract

Every Dedekind complete Riesz space X has a unique sup-completion X^{s}, which is a Dedekind complete lattice cone. This paper aims to present a systematic study this cone by extending several known results to general setting, proving new results and, in particular, introducing for elements of X^{s} finite and infinite parts. This enuables us to get a satisfactory abstract formulation of some classical results in the setting of Riesz spaces. We prove, in pareticular, a Riesz space version of Borel-Cantelli Lemma and present some applications to it.29*