Tests of Existence of Generalized Riemann Integrals

Abstract

1. The Perron integral 1(f) and, equivalently, the restricted Denjoy integral [13, 201, 241, 247]_were given an equivalent definition, many years later, by J. Kurzweil and R. Henstock. This definition is merely a quite simple variation of a familiar definition of the Riemann integral. The terms generalized Riemann integral (integrable), abbreviated here GRI, refer to 1(f) as defined by this variation. Thus GRI is at the same time very elementary but more powerful than Lebesgue and includes as special cases the Riemann, improper Riemann, Lebesgue and other integrals. It seems very sensible to make GRI the standard integral of the working analyst: [3]_is a textbook essentially doing this. (It uses, however, instead of GRI, the term gauge integral). [10] is a Carus Monograph on the subject, while [5, 6, 8, 12] are more technical monographs on GRI using for it also other names. The articles [2] and [9] introduce GRI and relate it to other integrals (e.g., improper Riemann). [2] contains also the simple proof of (1) below.