Perron-type integration on \(n\)-dimensional intervals and its properties. (English) Zbl 0832.26009

The paper introduces the \(\rho\)-integral over an interval \(I\) of \(\mathbb{R}^n\), which is a nonabsolutely convergent integral defined as a suitable limit of Riemann sums and using a more general regularity concept than other integrals of this type. Various properties, including a Saks-Henstock lemma and a convergence theorem, are proved, as well as continuity and differentiability properties of the primitives of such functions. A descriptive characterization of the integral is stated and proved. Finally, the class of strongly \(\rho\)-integrable functions, for which an approximation property through step functions is assumed, is introduced and compared with the \(\rho\)-integral.


26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
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