Ideal convergent subsequences and rearrangements for divergent sequences of functions

Abstract

Let $\I$ be an ideal on $\mathbb{N}$ 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 $\{0,1{\}}^{\mathbb{N}}$. Here we assume additionally that $\mathcal{I}$ has property (G).