Unbounded order convergence and application to martingales without probability

A net (xα)α∈Γ(xα)α∈Γ in a vector lattice X is unbounded order convergent (uo-convergent) to x   if |xα−x|∧y→o0 for each y∈X+y∈X+, and is unbounded order Cauchy (uo-Cauchy) if the net (xα−xα′)Γ×Γ(xα−xα′)Γ×Γ 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 L1(Ω;F)L1(Ω;F) is almost surely uo-Cauchy in F, where F is an order continuous Banach lattice with a weak unit.