Unbounded Order Convergence and Application to Martingales without Probability

2013/06/11 by Gao, Niushan, Xanthos, Foivos · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1306.2563

Abstract

A net (x_α)_{α∈ Γ} in a vector lattice X is unbounded order convergent (uo-convergent) to x if |x_α-x| \wedge y \xrightarrowo 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 Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in L₁(Ω;F) is almost surely uo-Cauchy in F, where F is an order continuous Banach lattice with a weak unit.

Cited by

• Bourgain-uo sequential completeness in vector lattices
• Strong completeness of Lp-type vector lattices

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