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
-integral (of Lusin type);
- the
-integral (of variational type);
- the
-integral (of Ward type);
- the
-integral (of Riemann type).
We prove that in certain conditions the integrals
and
are equivalent (it is shown that the first integral satisfies a Saks-Henstock type lemma).
For the
-integral we only show that it satisfies a quasi Saks Henstock type lemma (see Lemma 7.4).
Finally, if
and
we obtain that the integrals
,
and ![\small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}\bigr ] \small \bigl [{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}\bigr ]](http://latex.codecogs.com/gif.latex?%5Csmall%20%5Cbigl%20%5B%7B%5Cmathcal%20S%7D_o%5E%2B%20%7B%5Cmathcal%20S%7D_o%5E-%20%7B%5Cmathcal%20D%7D%5Cbigr%20%5D)
are equivalent.
In fact the
-integral is exactly the wide Denjoy integral.
But the equivalence of the three integrals above with the
-integral follows only if we assume the additional condition that the primitives of the
-integral are continuous (see Theorem 11.1)