Unbounded Order Convergence and Application to Martingales without Probability

A net $(x_{\alpha }{)}_{\alpha \in \mathrm{\Gamma }}$ in a vector lattice $X$ is unbounded order convergent (uo-convergent) to $x$ if $|x_\alpha-x| \wedge y \xrightarrow{o} 0$foreach$y \in X_+$, and is unbounded order Cauchy (uo-Cauchy) if the net $(x_{\alpha }-x_{{\alpha }^{\mathrm{\prime }}}{)}_{\mathrm{\Gamma }\times \mathrm{\Gamma }}$ 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 $L_1(\mathrm{\Omega };F)$ is almost surely uo-Cauchy in $F$, where $F$ is an order continuous Banach lattice with a weak unit.