Unbounded order convergence and application to martingales without probability

A net ( x α ) α ∈ Γ in a vector lattice X is unbounded order convergent (uo-convergent) to x if | x α − x | ∧ y → o 0 for each y ∈ X + , and is unbounded order Cauchy (uo-Cauchy) if the net ( x α − x α ′ ) 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 Doobʹs submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in L 1 ( Ω ; F ) is almost surely uo-Cauchy in F, where F is an order continuous Banach lattice with a weak unit.