The revival of the Riemannian approach to integration. (English) Zbl 1052.26008

Ciesielski, Zbigniew (ed.) et al., Orlicz centenary volume. Proceedings of the conferences ‘The Władysław Orlicz centenary conference’ and ‘Function spaces VII’, Poznań, Poland, July 21–25, 2003. Volume I: Plenary lectures. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 64, 147-158 (2004).
From the text: “Riemannian sums were used in a definition of a nonabsolutely convergent integration on \(\mathbb{R}\) almost fifty years ago and it was proved that the new integration is equivalent to Perron integration. Therefore it looked like there appeared just a new approach to integration. But the Riemannian approach proved to be very flexible: (i) it is the basis of numerous integrations of real functions of a real variable, (ii) it opens new ways in integration of functions of several variables, (iii) it is well applicable to integration of vector valued functions, (iv) it admits a great degree of abstraction. This paper is concentrated on (i) with a special accent on structures (convergence, topology) on the space of integrable functions. Section 5 contains a brief information concerning (ii).”
For the entire collection see [Zbl 1050.46003].


26A39 Denjoy and Perron integrals, other special integrals
26B99 Functions of several variables