A Classification of Order Convergence via a Transfinite Fatou Hierarchy

Abstract

We investigate the descriptive complexity of order convergence in separable Banach lattices. While uniform convergence is Borel and $\sigma$-order convergence is known to be ${\bf \Delta}^1_2$, it is unclear in general when $\sigma$-order convergence is analytic. We introduce a transfinite hierarchy of weakenings of the classical Fatou property, indexed by countable ordinals, and show that it provides a complete structural classification of this definability problem. For a separable Banach lattice $X$, we prove that the following are equivalent: (i) the set of decreasing positive sequences with infimum zero is Borel; (ii) $\sigma$-order convergence is analytic; and (iii) $X$ satisfies the $\alpha$-Fatou property for some countable ordinal $\alpha$. We further establish that the hierarchy is proper: for every countable ordinal $\alpha$ there exists a separable Banach lattice with a countable $\pi$-basis that fails to be $\alpha$-Fatou, but is $\beta$-Fatou for some $\beta>\alpha$. Thus the Borel definability of order convergence is governed by a canonical ordinal invariant intrinsic to the lattice, and the descriptive complexity can be arbitrarily high below $\omega_1$. These results identify projective complexity as a genuine structural invariant in Banach lattice theory.