Ideal convergent subsequences and rearrangements for divergent sequences of functions

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 $(X,μ)$ is a $σ$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\I$-divergent $μ$-almost everywhere. Then the set of subsequences of $(f_n)$, $\I$-divergent $μ$-almost everywhere, is of full product measure on $\{ 0,1\}^\N$. Here we assume additionally that $\mathcal I$ has property (G).

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