On \(\mathcal I\)-asymmetry (Q1852398)

The author presents a definition and some properties of a category analogue to the one-side density points. The notion of \({\mathcal I}\)-density and some related results can be found, for instance, in a survey article by \textit{W. Wilczyński} [Real Anal. Exch. 10(1984/85), 241-265 (1985; Zbl 0593.26008)]. Let \({\mathcal S}\) be the \(\sigma\)-algebra of all sets having the Baire property in \(\mathbb {R}\) and \({\mathcal I}\) in \({\mathcal S}\) the \(\sigma\)-ideal of all sets of the first category in \(\mathbb{R}\). A sequence \(\{f_n\}^{\infty}_{n=1}\) of \({\mathcal S}\)-measurable functions converges with respect to \({\mathcal I}\) to an \({\mathcal S}\)-measurable function \(f\) iff for every subsequence \(\{f_{n_m}\}^{\infty}_{m=1}\) of \(\{f_n\}^{\infty}_{n=1}\) there exists a subsequence \(\{f_{n_{m_p}}\}^{\infty}_{p=1}\) of \(\{f_{n_m}\}^{\infty}_{m=1}\) for which the following statement holds: \(\{x\in \mathbb{R}:\lim_{p\to \infty}f_{n_{m_p}}(x)\neq f(x)\}\in {\mathcal I}.\) Then a point \(a\) of \(\mathbb{R}\) is a right \({\mathcal I}\)-density point of a set \(A\in {\mathcal S}\) iff the sequence \(\{\chi_{(n(A-a))\cap [0,1]}\}^{\infty}_{n=1}\) converges with respect to \({\mathcal I}\) to \(\chi_{[0,1]}\). On an analogous way are defined the left \({\mathcal I}\)-density points. Based on those notions can be defined the \({\mathcal I}\)-asymmetry set and the strong \({\mathcal I}\)-asymmetry set of a Baire function \(f: \mathbb{R}\to \mathbb{R}\). Some properties of those \({\mathcal I}\)-asymmetry sets are obtained in the paper. It is proved, for instance, that the \({\mathcal I}\)-asymmetry set of a function can be of the cardinality of the continuum whereas the strong \({\mathcal I}\)-asymmetry set is countable.