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A descriptive definition of some multidimensional gauge integrals. (English) Zbl 0852.26010
The following multidimensional integral, motivated by earlier work of Mawhin, Jarník, Kurzweil, Schwabik, Pfeffer and Nonnenmacher, is introduced in this paper. A function \(f: I\to \mathbb{R}\) is said to be \(M_0\)-integrable with integral \(c\) if for any \(\varepsilon> 0\) and any \(K> 0\) there exists a gauge \(\delta: I\to \mathbb{R}_+\) with the property \(|S(f, \Pi, I)- c|\leq \varepsilon\) for every \(\delta\)-fine P-partition \(\Pi= \{(x^1, I^1),\dots, (x^s, I^s)\}\) with \(\Sigma_0(\Pi):= \sum^s_{i= 1} l(J_i)^n\leq K\). For an interval \(J\), \(l(J)\) is the length of the larger side. It is then proved that if \(F\) is an additive interval function which is differentiable on the interior of an interval containing an interval \(I\), then its derivative \(F'\) is \(M_0\)-integrable on \(I\) and \(\int_I F'= F(I)\). Other standard properties are also proved for the \(M_0\)-integral. The space of \(M_0\)-integrable functions contains that of \(M_1\)-integrable functions in the sense of Jarník, Kurzweil and Schwabik, and is contained in the space of functions integrable in the sense of Pfeffer.
The paper also introduces outer measures associated to various integration processes and characterizes the corresponding indefinite integrals. Some open questions are indicated.

MSC:
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
26A39 Denjoy and Perron integrals, other special integrals
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References:
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