## 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.

### MSC:

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

 [1] Chew T.S.: On the equivalence of Henstock-Kurzweil and restricted Denjoy integrals in $$R^n$$. Real Analysis Exchange 15 (1989-90), 259-268. · Zbl 0704.26016 [2] Jarník J., Kurzweil J., Schwabik Š.: On Mawhin’s approach to multiple nonabsolutely convergent integral. Časopis pěst. mat. 108 (1983), 356-380. · Zbl 0555.26004 [3] Kurzweil J., Jarník J.: Equiintegrability and controlled convergence of Perron-type integrable functions. Real Analysis Exchange 17 (1991-92), 110-139. · Zbl 0754.26003 [4] Kurzweil J., Jarník J.: Differentiability and integrability in $$n$$ dimensions with respect to $$\alpha$$-regular intervals. Resultate der Mathematik 21 (1992), 138-151. · Zbl 0764.28005 [5] Mawhin J.: Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields. Czechoslovak Math. J. 31(106) (1981), 614-632. · Zbl 0562.26004 [6] Pfeffer W. F.: A Riemann type integration and the fundamental theorem of calculus. Rendiconti Circ. Mat. Palermo, Ser. II 36 (1987), 482-506. · Zbl 0669.26007
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