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Article 43

A study of some general integrals which contain the wide Denjoy integral

Real Analysis Exchange 26 (2000/2001), no. 1, 51-100.

 Mathematical reviews subject classification: 26A39; 26A42; 26A45.

 

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)