Jarník, Jiří; Kurzweil, Jaroslav Perron-type integration on \(n\)-dimensional intervals and its properties. (English) Zbl 0832.26009 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) Cited in 1 ReviewCited in 8 Documents 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 PDF BibTeX XML Cite \textit{J. Jarník} and \textit{J. Kurzweil}, Czech. Math. J. 45, No. 1, 79--106 (1995; Zbl 0832.26009) Full Text: EuDML OpenURL References: [1] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechosl. Math. J. 7 (1957), 418-449. · Zbl 0090.30002 [2] J. Kurzweil: Nichtabsolutkonvergente Integrale, Teubner Texte zur Mathematik, 25. Teubner, Leipzig, 1984. [3] J. Kurzweil, J. Jarník: Equiintegrability and controlled convergence of Perron-type integrable functions. Real. Anal. Exchange 17 (1991-92), 110-139. · Zbl 0754.26003 [4] J. Kurzweil, J. Jarník: Differentiability and integrability in \(n\) dimensions with respect to \(\alpha \)-regular intervals. Results in Mathematik 21 (1992), 138-151. · Zbl 0764.28005 [5] J. Kurzweil, J. Jarník: Generalized multidimensional Perron integral involving a new regularity condition. Results in Mathematics 23 (1993), 263-273. · Zbl 0782.26003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.