Burkholder Theorem in Riesz spaces

TL;DR

This paper extends Burkholder's martingale convergence theorem to Riesz spaces, utilizing vector lattice analogues and new characterizations to broaden the theorem's applicability in abstract settings.

Contribution

It provides a vector lattice version of Burkholder's theorem, incorporating recent advances in sample function theorems and element characterization in Riesz spaces.

Findings

Established a vector lattice analogue of Burkholder's theorem

Developed a new characterization of elements in the sup-completion of Riesz spaces

Extended martingale convergence results to abstract vector lattice settings

Abstract

The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice which do not belong to the space.