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

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).