On absolute continuity
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 \( \overline DF(x) = \sup_{0< \eta< 1/2m}\inf _{\delta>0}\sup_B{F(B)\over |B|}\), where \(B\) is an \(\eta\)-regular figure, \( x\in B\), and the diameter of \(B\) is less than \(\delta\). This result extends to all \(m\) the result that \textit{B. Bongiorno, L. Di Piazza} and \textit{V. Skvortsov} [Real Anal. Exch. 21, No. 2, 656-663 (1995; Zbl 0879.26026)] proved, in a different manner, in the case \(m=1\). An elegant corollary is: if \( F\) is a charge in a figure \(A\) then (i) if \(V_*F\) is AC then \(F\) is the indefinite R-integral of \(DF\); (ii) if \(V_*F\) is AC and finite then \(F\) is the indefinite Lebesgue integral of \(DF\).
scientific article; zbMATH DE number 1217000