Unbounded Norm Convergence in Banach Lattices

A net $(x_α)$ in a vector lattice $X$ is unbounded order convergent to $x \in X$ if $\lvert x_α- x\rvert \wedge u$ 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 $(x_α)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\lVert\lvert x_α- x\rvert \wedge u\rVert\to 0$ 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.

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