A net $$(x_\alpha )$$ ( x α ) in a vector lattice X is unbounded order convergent to $$x \in X$$ x ∈ X if $$|x_\alpha - x| \wedge u$$ | x α - x | ∧ u converges to 0 in order for all $$u\in X_+$$ 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 $$(x_\alpha )$$ ( x α ) in a Banach lattice X is unbounded norm convergent to x if [InlineEquation not available: see fulltext.] for all $$u\in X_+$$ 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.
Positivity – Springer Journals
Published: Sep 30, 2016
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