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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.

MSC:

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