Unbounded Order Convergence and Application to Martingales without Probability

Abstract

A net $(x_{\alpha }{)}_{\alpha \in \Gamma }$ in a vector lattice $X$ is unbounded order convergent (uo-convergent) to $x$ if $|x_{\alpha }-x|\wedge y\stackrel{o}{\to }0$ for each $y\in X_{+}$, and is unbounded order Cauchy (uo-Cauchy) if the net $(x_{\alpha }-x_{{\alpha }^{\prime }}{)}_{\Gamma \times \Gamma }$ is uo-convergent to 0. In the first part of this article, we study uo-convergent and uo-Cauchy nets in Banach lattices and use them to characterize Banach lattices with the positive Schur property and KB-spaces. In the second part, we use the concept of uo-Cauchy sequences to extend Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in $L_1(\Omega ;F)$ is almost surely uo-Cauchy in $F$, where $F$ is an order continuous Banach lattice with a weak unit.

Cited by in corpus (2)

• Unbounded order convergence in dual spaces
• The countable sup property for lattices of continuous functions