On convergence theorems for AP integrals (Q1890597)

Let \(x\in [a, b]\), \(U_ x\) a measurable subset of \([a, b]\) with \(x\in U_ x\) and having density 1 at \(x\). The collection of all interval point pairs \(\{[u, v], x\}\) with \(u, v\in U_ x\), \(u\leq x\leq v\), is called an AFC of \([a, b]\), \(\Delta\) say. A sub-collection \(D\) of \(\Delta\), \(\{[x_{i- 1}, x_ i], \xi_ i\}\) with \(a= x_ 0< x_ 1<\cdots< x_ n= b\), is called a \(\Delta\)-partition of \([a, b]\); and a subset of \(D\) is called a partial \(\Delta\)-partition; if in addition the points of a partial \(\Delta\)-partition, the \(\xi_ i\), lie in a set \(X\) we call it a partial \(\Delta\)-partition on \(X\). Now if \(F: [a, b]\to \mathbb{R}\), \(X\subseteq [a, b]\) then \(F\in \text{AC}^{**}_{ap}(X)\) if: \(\forall \varepsilon\) \(\exists \Delta\) and an \(\eta> 0\) such that for any two partial \(\Delta\)-partitions on \(X\), \(D_ 1\), \(D_ 2\) of \([a, b]\), \[ (D_ 1\backslash D_ 2) \sum | v- u|< \eta\Rightarrow (D_ 1\backslash D_ 2) \sum | F(v)- F(u)|< \varepsilon. \] Here, the sums are over all interval-point pairs of \(D_ 1\backslash D_ 2\), where \(D_ 1\backslash D_ 2\) is the collection of intervals in \(E_ 1\backslash E_ 2\), where \(E_ 1\) is the union of the intervals in \(D_ 1\), and \(E_ 2\) the union of those in \(D_ 2\). If we taken \(D_ 2= \emptyset\) in this definition we say that \(F\in \text{AC}^*_{ap}(X)\). Clearly \(\text{AC}^{**}_{ap}(X)\subseteq \text{AC}^*_{ap}(X)\), although \(\text{ACG}^{**}_{ap}(X)= \text{AG}^*_{ap}(X)\), by Theorem 2 of this paper. The main result is the following convergence theorem for the approximately continuous Perron integral, the AP integral. \textit{Hypotheses}: \(f_ n\in \text{AP}\), \(n\in \mathbb{N}\); \(F_ n\) is the AP-primitive of \(f_ n\), \(n\in \mathbb{N}\); \(f_ n\to f\) a.e. in \([a, b]\); \(\{F_ n\}\) is uniformly \(\text{ACG}^{**}_{ap}\) on \([a, b]\). \textit{Conclusions}: \(f\) is AP-integrable and \(\int^ b_ a f= \lim_{n\to \infty} \int^ b_ a f_ n\). The authors use a clever argument, not using the uniform \(\text{ACG}^{**}_{ap}\) concept, uniform \(\text{ACG}^*_{ap}\) will do, to reduce the above result to proving a convergence theorem with the extra hypothesis: \(F_ n\to F\). This is a result proved earlier [\textit{Y.-J. Lin}, Real Anal. Exch. 19, No. 1, 155-164 (1994; Zbl 0813.26002)] by a different method based on the one used for a similar result for the Henstock-Kurzweil integral.