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 \(|\sum_{i=1}^nf(x_i)|A_i|-F(A_i)|0}\sup_B{F(B)\over |B|}\), 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 \(DF\).