Equivalent definitions of regular generalized Perron integral. (English) Zbl 0782.26004

In 1981, the reviewer introduced a generalization of the multidimensional Kurzweil-Henstock integral which integrates the divergence of every differentiable vector field over an \(n\)-dimensional interval. However, such an integral was not additive. To overcome this difficulty, Pfeffer introduced in 1986 an additive integral providing a similar divergence theorem. The price was a more complicated definition than the reviewer’s one. In this interesting paper, the authors show that a function \(f\) is integrable in Pfeffer’s sense over an \(n\)-dimensional interval \(I\) if and only if its extension by zero to \(\mathbb{R}^ n\) is integrable in the reviewer’s sense over an \(n\)-dimensional interval \(J\) such that \(I\subset \text{int }J\). To prove this result, the authors give various interesting equivalent definitions of the Pfeffer’s integral.


26B15 Integration of real functions of several variables: length, area, volume
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