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

Jarník, Jiří; Kurzweil, Jaroslav

Czech. Math. J. 45, No. 1, 79-106 (1995).

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.

Reviewer: J.Mawhin (Louvain-La-Neuve)

MSC:

26B20

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

Keywords:

Kurzweil-Henstock integrals; Perron-type integration; nonabsolutely convergent integral; Saks-Henstock lemma; convergence theorem

References:

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