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 
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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Acknowledgments
We would like to thank Niushan Gao for valuable discussions.
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V. G. Troitsky was supported by an NSERC Grant.
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Deng, Y., O’Brien, M. & Troitsky, V.G. Unbounded norm convergence in Banach lattices. Positivity 21, 963–974 (2017). https://doi.org/10.1007/s11117-016-0446-9
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DOI: https://doi.org/10.1007/s11117-016-0446-9