Spaces of regular abstract martingales

TL;DR

This paper investigates the structure of regular martingale spaces on vector and Banach lattices, revealing conditions under which these spaces form vector or Banach lattices, thus deepening understanding of their lattice properties.

Contribution

It demonstrates that regular martingale spaces may not always be vector lattices, but become so under order completeness, and also explores their Banach lattice structure.

Findings

Regular martingale spaces are not necessarily vector lattices.

Order completeness of the underlying space ensures the regular martingale space is a vector lattice.

Under certain conditions, these spaces form Banach lattices with the regular norm.

Abstract

In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm.