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.
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Under analytic determinacy, σ-order bases in Banach lattices are uniform and hence Schauder, order and σ-order bases coincide, a Banach space exists with a filter Schauder basis but no ordinary one, and analytic-filter bases reduce to Borel-filter bases.
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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.
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Coordinate systems in Banach spaces and lattices
Under analytic determinacy, σ-order bases in Banach lattices are uniform and hence Schauder, order and σ-order bases coincide, a Banach space exists with a filter Schauder basis but no ordinary one, and analytic-filter bases reduce to Borel-filter bases.