Egorov ideals

TL;DR

The paper investigates Egorov ideals on $\omega$, examining when the ideal-version of Egorov's theorem holds for $\mathcal{I}$-pointwise and $\mathcal{I}$-uniform convergence on $[0,1]$. It proves a sharp dichotomy for non-pathological $\Sigma^0_2$ ideals: Egorov iff the ideal is countably generated, which implies only three such ideals are possible up to isomorphism, namely $\mathrm{Fin}$, $\mathrm{Fin}\oplus\mathcal{P}(\omega)$, and $\mathrm{Fin}\otimes\{\emptyset\}$. The work also constructs $2^{\omega}$ pairwise non-isomorphic Borel Egorov ideals and analyzes how Egorov-ness behaves under products, sums, and intersections of ideals, including Rudin-Keisler order considerations for pathological cases. It further shows that certain well-known pathological $\Sigma^0_2$ ideals (e.g., Mazur's and Solecki's) are non-Egorov, while providing a framework to generate large families of Egorov ideals via combinatorial constructions. Overall, the results illuminate the landscape of ideal convergence and provide exact structural characterizations in key descriptive set-theoretic classes.

Abstract

We study Egorov ideals, that is ideals on $ω$ for which the Egorov's theorem for ideal versions of pointwise and uniform convergences holds. We show that a non-pathological $\bf{Σ^0_2}$ ideal is Egorov if and only if it is countably generated. In particular, up to isomorphism, there are only three non-pathological $\bf{Σ^0_2}$ Egorov ideals. On the other hand, we construct $2^ω$ pairwise non-isomorphic Borel Egorov ideals. Moreover, we characterize when a product of ideals is Egorov.

Theorems & Definitions (59)

• Definition 1.1
• Example 2.1
• Remark 3.1
• Proposition 3.2
• proof
• Theorem 3.3: KadetsLeonov
• Theorem 3.4
• proof
• Definition 4.1
• Lemma 4.2