## 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.

### MSC:

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

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