arXiv subjects
Publications and source records attributed to Artur Wachowicz.
We prove that the lower density operator associated with the Baire category density points in the real line has Borel values of class $\pmb Π^0_3$ which is analogous to the measure case. We also introduce the notion of the Baire category density point of a subset with the Baire property of the Cantor space, and we prove that it generates a lower density operator with Borel values of class $\pmb Π^0_3$.
Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and $\sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(\mathcal{I}):=\left\{t \in \{0,1\}^{\mathbb{N}} \colon \sum_n t(n)x_n \textrm{ is } \mathcal{I}\textrm{-convergent}\right\}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure $λ$ on $\{0,1\}^{\mathbb{N}}$ is considered. If $\mathcal{I}$ is analytic or coanalytic, and $\sum_n x_n$ is $\mathcal{I}$-divergent, then $λ(A(\mathcal{I}))=0$ which extends the theorem of Dindoš, Šalát and Toma. Generalizing one of their examples, we show that, for every ideal $\mathcal{I}$ on $\mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $λ(A(Fin))=0$ and $λ(A(\mathcal{I}))=1$.
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).