A Classification of Order Convergence via a Transfinite Fatou Hierarchy
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arXiv
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| Natura: | Preprint |
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2026
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| author | Avilés, Antonio Rosendal, Christian Taylor, Mitchell A. Tradacete, Pedro |
| author_facet | Avilés, Antonio Rosendal, Christian Taylor, Mitchell A. Tradacete, Pedro |
| contents | We investigate the descriptive complexity of order convergence in separable Banach lattices. While uniform convergence is Borel and $σ$-order convergence is known to be ${\bf Δ}^1_2$, it is unclear in general when $σ$-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) $σ$-order convergence is analytic; and (iii) $X$ satisfies the $α$-Fatou property for some countable ordinal $α$.
We further establish that the hierarchy is proper: for every countable ordinal $α$ there exists a separable Banach lattice with a countable $π$-basis that fails to be $α$-Fatou, but is $β$-Fatou for some $β>α$. 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 $ω_1$. These results identify projective complexity as a genuine structural invariant in Banach lattice theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02588 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Classification of Order Convergence via a Transfinite Fatou Hierarchy Avilés, Antonio Rosendal, Christian Taylor, Mitchell A. Tradacete, Pedro Functional Analysis 46B42, 46B15, 03E15, 46H40, 54A20 We investigate the descriptive complexity of order convergence in separable Banach lattices. While uniform convergence is Borel and $σ$-order convergence is known to be ${\bf Δ}^1_2$, it is unclear in general when $σ$-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) $σ$-order convergence is analytic; and (iii) $X$ satisfies the $α$-Fatou property for some countable ordinal $α$. We further establish that the hierarchy is proper: for every countable ordinal $α$ there exists a separable Banach lattice with a countable $π$-basis that fails to be $α$-Fatou, but is $β$-Fatou for some $β>α$. 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 $ω_1$. These results identify projective complexity as a genuine structural invariant in Banach lattice theory. |
| title | A Classification of Order Convergence via a Transfinite Fatou Hierarchy |
| topic | Functional Analysis 46B42, 46B15, 03E15, 46H40, 54A20 |
| url | https://arxiv.org/abs/2604.02588 |