On absolute continuity
type: Item
instance of: scholarly article
title: On absolute continuity
zbMATH Open document ID: 0931.26005
author: Zoltán Buczolich
author: Washek F. Pfeffer
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-09-02T00:00:00Z
review text: The family \({\mathcal F}\) of all figures in \(\mathbb R^m\), finite unions of non-degenerate compact intervals, is topologized with a Hausdorff topology \( \tau\) [ see \textit{L. C. Evans} and \textit{R. F. Gariepy}: ``Measure theory and fine properties of functions'' (1992; Zbl 0804.28001)] and an additive \( \tau\)-continuous real-valued function on \({\mathcal F}\) is called a charge; examples of charges are the indefinite Lebesgue integral of a locally integrable function, and also the flux of a continuous vector field [see \textit{W. F. Pfeffer}: ``The Riemann approach to integration: local geometric theory'' (1993; Zbl 0804.26005)]. A charge \(F\) is said to be a charge in a figure \(A\) if for all figures \(B, F(B) = F(\overline {\text{ int}A\cap\text{ int} B})\). A function \(f\) on a figure \(A\) is R-integrable if there is a charge \(F\) in \(A\) such that for all \(\varepsilon>0\) there is a gage \(\delta\) on \(A\) such that for all \(\varepsilon\)-regular, \(\delta\)-fine partitions of \(A\), \(\{(A_,x_i); 1\leq i\leq n\}\), we have that \(|\sum_{i=1}^nf(x_i)|A_i|-F(A_i)|< \varepsilon\); the terminology is standard in the theory of generalized integrals; see the previous reference. The charge \(F\) is then called the indefinite R-integral of \(f\). If \(F\) is a function on \({\mathcal F}\) then the critical variation of \(F\) on a set \(E\) is \(V_*(E) = \sup_{1< \eta< 1/2m}\inf_ \delta \sup_{ \varpi} \sum_{i=1}^p|F(A_i)|\), where \( \varpi= \{(A_,x_i)\); \(1\leq i\leq p\}\) is an \(\eta\)-regular \(\delta\)-fine partition where \(\delta\) is gage on \(E\). The main result is : a charge \(F\) in a figure \(A\) is an indefinite R-integral of an \(f\) on \(A\) if and only if \(V_*F\) is AC; and then \(DF= f\) a.e. in \(A\), \(f\) is measurable and \(V_*(E) = \int_E |f|\), for every measurable \(E\subseteq A\). Here, as is usual, \(DF\) is the finite common value of \( \underline D F\) and \( \overline DF\), where \( \overline DF(x) = \sup_{0< \eta< 1/…
Mathematics Subject Classification ID: 26B30
Mathematics Subject Classification ID: 26A39
zbMATH DE Number: 1217000
zbMATH Keywords: variation absolute continuity
zbMATH Keywords: generalized Riemann integral
zbMATH Keywords: Lebesgue integral
zbMATH Keywords: charges
reviewed by: P. S. Bullen
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5804
OpenAlex ID: W1990545367
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DOI: 10.1006/JMAA.1997.5804
description: scientific article; zbMATH DE number 1217000