A Study of Some General Integrals that Contains the Wide Denjoy Integral

Abstract

In this paper, using Thomson's local systems, we introduce some very general integrals, each containing the wide Denjoy integral: the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal D}]$-integral (of Lusin type); the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal V}]$-integral (of variational type); the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal W}]$-integral (of Ward type); the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal R}]$-integral (of \linebreak Riemann type); We prove that in certain conditions the integrals $[{\mathcal S}_1 \!{\mathcal S}_2 {\mathcal V}]$ and $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal W}]$ are equivalent (it is shown that the first integral satisfies a Saks-Henstock type lemma). For the $[{\mathcal S}_1{\mathcal S}_2{\mathcal R}]$-integral we only show that it satisfies a quasi Saks Henstock type lemma (see Lemma 7.4). Finally, if ${\mathcal S}_1 = {\mathcal S}_o^+$ and ${\mathcal S}_2 = {\mathcal S}_o^-$ we obtain that the integrals $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal V}]$, $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal W}]$ and $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}]$ are equivalent (in fact the $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}]$-integral is exactly the wide Denjoy integral). But the equivalence of the three integrals with the $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal R}]$-integral follows only if we assume the additional condition that the primitives of the $[{\mathcal S}_o^+{\mathcal S}_o^- {\mathcal R}]$-integral are continuous (see Theorem11.1).