A category analogue of the density topology, approximate continuity and the approximate derivative
type: Item
instance of: scholarly article
title: A category analogue of the density topology, approximate continuity and the approximate derivative
zbMATH Open document ID: 0593.26008
published in: Real Analysis Exchange
publication date: 1985-01-01T00:00:00Z
review text: 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 \((n(A-a))\cap (-1,1).\)
reviewed by: Ladislav Mišík
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 3953132
zbMATH Keywords: approximate continuity
zbMATH Keywords: approximate derivative
zbMATH Keywords: \(\sigma\)-algebra of all
zbMATH Keywords: subsets of R having the Baire property
zbMATH Keywords: category analogue of the density point
zbMATH Keywords: \(\sigma\)-ideal
zbMATH Keywords: first category
author: Władysław Wilczyński
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3953132