Ideal convergent subsequences and rearrangements for divergent sequences of functions

TL;DR

This paper studies the Baire category and measure-theoretic properties of subsequences and rearrangements of divergent function sequences, generalizing Kallman's theorem within the framework of ideals on natural numbers.

Contribution

It extends existing results by analyzing $ ext{I}$-convergence and divergence of function sequences under ideals with property (G), covering both Baire category and measure contexts.

Findings

$ ext{I}$-convergent subsequences are meager in the Baire category sense.

The set of $ ext{I}$-divergent subsequences has full measure under certain conditions.

Generalization of Kallman's theorem to broader classes of ideals and function spaces.

Abstract

Let $\I$ be an ideal on $\N$ which is either analytic or coanalytic. Assume that $(f_n)$ is a sequence of functions with the Baire property from a Polish space $X$ into a complete metric space $Z$, which is divergent on a comeager set. We investigate the Baire category of $\I$-convergent subsequences and rearrangements of $(f_n)$. Our result generalizes a theorem of Kallman. A similar theorem for subsequences is obtained if $(X,\mu)$ is a $\sigma$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\I$-divergent $\mu$-almost everywhere. Then the set of subsequences of $(f_n)$, $\I$-divergent $\mu$-almost everywhere, is of full product measure on $\{ 0,1\}^\N$. Here we assume additionally that $\mathcal I$ has property (G).