Unbounded Norm Convergence in Banach Lattices

Abstract

A net $(x_{\alpha })$ in a vector lattice $X$ is unbounded order convergent to $x\in X$ if $|x_{\alpha }-x|\wedge u$ converges to $0$ in order for all $u\in X_{+}$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_{\alpha })$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\Vert |x_{\alpha }-x|\wedge u\Vert \to 0$ for all $u\in X_{+}$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.