A category analogue of the density topology, approximate continuity and the approximate derivative
The paper presents a definition of a category analogue of the density point and a survey of almost all results connected with this notion which were obtained in Lodź during the last three years and which were or will be published in separate papers. The basic systems are: the \(\sigma\)-algebra \({\mathcal S}\) of all sets having the Baire property in R and the \(\sigma\)-ideal \({\mathcal I}\) in \({\mathcal S}\) of all sets of the first category in R. The well-known theorem of F. Riesz on the convergence in measure of a sequence of measurable functions to a measurable function is the basic idea of the definition of the convergence with respect to \({\mathcal I}\) of a sequence of \({\mathcal S}\)-measurable functions to an \({\mathcal S}\)-measurable function. There is the basic definition: 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 holds: \(\{x\in R:\lim_{p\to \infty}f_{n_{m_ p}}(x)\neq f(x)\}\in {\mathcal I}.\) Then a point a of R is an \({\mathcal I}\)-density point of a set \(A\in {\mathcal S}\) iff the sequence \(\{c_{(n(A-a))\cap (-1,1)}\}^{\infty}_{n=1}\) converges with respect to \({\mathcal I}\) to the zero function, where \(c_{(n(A-a))\cap (-1,1)}\) is the characteristic function of the set \((n(A-a))\cap (-1,1).\)
scientific article; zbMATH DE number 3953132