The generalized Riemann-Stieltjes integral. (English) Zbl 0879.26043

A Stieltjes type integral \(\int_Af dG\) of a function \(f\) with respect to a charge \(G\) in the bounded \(BV\) set \(A\subset \mathbb R^m\) is defined in the paper via Riemann type integral sums and the concept of the indefinite integral. Some specific results concerning the concept are presented. E.g., it is shown that a function which is integrable with respect to all charges is essentially bounded and \(BV\) (\(BV\) means that the function is Lebesgue integrable over \(\mathbb R^m\) and its distributional derivative is a vector measure in \(\mathbb R^m\) whose variation is finite). Some other interesting results are presented in the direction to describe the linear space of all functions that are integrable with respect to every charge.


26B30 Absolutely continuous real functions of several variables, functions of bounded variation
26A39 Denjoy and Perron integrals, other special integrals