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Deng, Y.; O’Brien, M.; Troitsky, V. G.
Positivity : an international journal devoted to the theory and applications of positivity in analysis, 09/2017, Volume: 21, Issue: 3Journal Article
A net ( x α ) in a vector lattice X is unbounded order convergent to x ∈ X if | x α - x | ∧ u converges to 0 in order for all u ∈ 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 for all u ∈ X + . We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.
