23 | 10 | 2019
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# A study of some general integrals which contain 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 $\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal D}\bigr ]$-integral (of Lusin type);
• the $\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal V}\bigr ]$-integral (of variational type);
• the $\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal W}\bigr ]$-integral (of Ward type);
• the $\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal R}\bigr ]$-integral (of Riemann type).

We prove that in certain conditions the integrals$\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal V}\bigr ]$ and $\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal W}\bigr ]$ are equivalent (it is shown that the first integral satisfies a Saks-Henstock type lemma).

For the$\small \bigl [{\mathcal S}_1 {\mathcal S}_2 {\mathcal R}\bigr ]$-integral we only show that it satisfies a quasi Saks Henstock type lemma (see Lemma 7.4).

Finally, if $\small{\mathcal S}_1 = {\mathcal S}_o^+$ and $\small{\mathcal S}_2 = {\mathcal S}_o^-$ we obtain that the integrals

$\small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^-{\mathcal V}\bigr ]$,    $\small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^-{\mathcal W}\bigr ]$    and   $\small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}\bigr ]$

are equivalent.

In fact the$\small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^-{\mathcal D}\bigr ]$-integral is exactly the wide Denjoy integral.

But the equivalence of the three integrals above with the$\small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal R}\bigr ]$-integral follows only if we assume the additional condition that the primitives of the$\small \bigl [{\mathcal S}_o^+{\mathcal S}_o^- {\mathcal R}\bigr ]$-integral are continuous  (see Theorem 11.1)

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