Unbounded order convergence and application to martingales without probability.

Abstract: A net in a vector lattice X is unbounded order convergent (uo-convergent) to x if for each , and is unbounded order Cauchy (uo-Cauchy) if the net 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 is almost surely uo-Cauchy in F, where F is an order continuous Banach lattice with a weak unit. [Copyright &y& Elsevier]