Ideal version of Egorov's theorem for analytic P-ideals

Abstract

We introduce the notion of equi-ideal convergence and use it to prove an ideal variant of Egorov's theorem. We also show that this variant usually cannot be strengthen to a direct analogue of Egorov's theorem.

A Generalized Egorov’s Statement for Ideals

“…, for any A, B ⊆ ω) and Exh(φ) = {A ⊆ ω∶ lim n→∞ φ(A ∖ n) = 0} (see also [4]). Fix a lower continuous submeasure φ such that I = Exh(φ).…”

“…equi-ideal convergence. And, for example, in the case of analytic P-ideal so called weak Egorov's Theorem for ideals (between equi-ideal and pointwise ideal convergence) was proved by N. Mrożek (see [4,Theorem 3.1]). Therefore, we ask whether in the case of an ideal and two notion of convergence for which the Egorov's theorem with measurability assumption holds, we can drop this assumption.…”

The Banach Principle for ideal convergence in the classical and noncommutative context

“…for which ideals) does I-Egorov convergence coincide with a.e. I-convergence has been recently addressed in [12].…”

I-selection principles for sequences of functions

“…In [4], Balcerzak et al proved an ideal version of Egorov's theorem for the ideal I d of statistical density zero sets. Recently, Mrożek [5] extended their result to all analytic P-ideals. Other ideals were considered by Kadets and Leonov in [6].…”