Approach to non-absolute integration by successive approximations (Q2758102)

The Denjoy, Perron and Henstock-Kurzweil integration processes generate equivalent integrals that extend the Lebesgue integral by allowing conditional convergence. The Denjoy integral is a countable extension of the Lebesgue integral on the real line. If \(f:[a,b]\to\mathbb R\) and \(f\) is Denjoy integrable then there exists a non-decreasing sequence of closed sets \(\{X_n\}\) such that \(\lim X_n=[a,b]\) and \(f\) is Lebesgue integrable on each \(X_n\). The original construction of the special Denjoy integral (1912) involved a transfinite limit of Lebesgue integrals. The Perron and Henstock-Kurzweil integrals accomplish the same, the former using major and minor functions, the latter using Riemann sums. This paper makes a survey of (mainly the author's contributions to) non-absolute integration defined using sequences of Lebesgue integrals. Functions mapping from the real line and \({\mathbb R}^n\) to the real line and functions mapping from the real line to ranked spaces or nuclear spaces are integrated with respect to Lebesgue measure. NEWLINENEWLINENEWLINEIn 1955 the author showed how to define the Denjoy integral via a countable process. Other highlights are Kunugi's use of ranked spaces and the author's use of nuclear spaces. NEWLINENEWLINENEWLINEThe paper is divided into ten sections.NEWLINENEWLINENEWLINE1. The Denjoy integral defined in countable process.NEWLINENEWLINENEWLINE2. Kunugi's idea for non-absolute integration.NEWLINENEWLINENEWLINE3. The \(A\)-integral and the special Denjoy integral defined by the procedure of completion.NEWLINENEWLINENEWLINE4. Basic principles of functional analysis in the space of special Denjoy integrals.NEWLINENEWLINENEWLINE5. Convergences.NEWLINENEWLINENEWLINE6. Multidimensional integrations (in connection with Fubini's theorem).NEWLINENEWLINENEWLINE7. Multidimensional integrations (in connection with divergence theorem).NEWLINENEWLINENEWLINE8. Multidimensional integrations (in connection with Riemann approach).NEWLINENEWLINENEWLINE9. Approximation theorems.NEWLINENEWLINENEWLINE10. Nuclear mapping and Riemann approach to vector valued integration.NEWLINENEWLINENEWLINEThe bibliography contains 67 items, 29 of which are by the author.