σ-order convergence is analytic in a separable Banach lattice exactly when the lattice satisfies the α-Fatou property for some countable ordinal α, and this hierarchy is proper at every level.
Taylor, and Pedro Tradacete,Coordinate systems in Banach spaces and lattices, Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure (to appear)
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A Classification of Order Convergence via a Transfinite Fatou Hierarchy
σ-order convergence is analytic in a separable Banach lattice exactly when the lattice satisfies the α-Fatou property for some countable ordinal α, and this hierarchy is proper at every level.