On a generalized dominated convergence theorem for the AP integral (Q1890598)

This paper contains a proof of the Harnack property for the AP integral and the following dominated convergence theorem for the same integral. \textit{Hypotheses}: \(f_ n\) is AP-integrable on \([a, b]\), \(n\in \mathbb{N}\); \(F_ n\) is the AP-primitive of \(f_ n\), \(n\in \mathbb{N}\); \(f_ n\to f\) a.e. in \([a, b]\); there are closed sets \(X_ k\), \(k\in \mathbb{N}\), with \(| [a, b]\backslash \bigcup_ k X_ k|= 0\) and such that for all \(n\in \mathbb{N}\), \(F_ n\in W(X_ k)\), \(k\in \mathbb{N}\); for each \(k\in \mathbb{N}\) there are \(g_ k\), \(h_ k\), L-integrable on \(X_ k\), such that for all \(n\in \mathbb{N}\), \(g_ k\leq f_ n\leq h_ k\) a.e. on \(X_ k\); for each \(k\in \mathbb{N}\), \(\sum_ j| \int_{I_{j, k}} f_ n|\) converges uniformly in \(n\), where \(I_{j, k}\) are the contiguous intervals of \(X_ k\); \(F_ n\to F\) on \([a, b]\); \(\{F_ n\}\) is uniformly ASL on \([a, b]\). \textit{Conclusions}: \(f\) is AP-integrable, with AP-primitive \(F\), and \(\int^ b_ a f= \lim_{n\to \infty} \int^ b_ a f_ n\). To say that \(F\in W(X)\), \(X\) closed, means: for all \(c,d\in X\) with \(]c, d[\cap X\neq \emptyset\), and for all \(\varepsilon> 0\) there is an AFC \(\Delta\) (see the preceding review) such that for all partial \(\Delta\)-partitions \(D\) on \(X\cap [c, d]\backslash H'\), \(H'\) the set of all points of density of \([c, d]\cap X\), we have \((D) \sum_{v\in ]c_ k, d_ k[} | F(v)- F(c_ k)|< \varepsilon\) and \((D) \sum_{u\in ]c_ k, d_ k[}| F(d_ k)- F(u)|< \varepsilon\). To say that \(F\) is ASL on \(X\) is to say that for every set of measure zero \(E\) and every \(\varepsilon> 0\) there is an AFC \(\Delta\) such that for any partial \(\Delta\)-partition, \(D\), on \(E\cap X\), \((D)\sum | F(v)- F(u)|< \varepsilon\). The author gives an example to show the power of his result. If \(F(x)= x^ 2\sin(1/ x^ 2)\), \(0< x\leq 1\), \(= 0\) otherwise and then if \(f_ n(x)= F'(x)\), \(n^{-1}\leq x\leq 1\), \(= 0\) otherwise, the above theorem applies to the sequence \(\{f_ n\}\), a sequence that is not dominated above or below by an AP-integrable function.