Unbounded norm convergence in Banach lattices

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.