On convergence theorems for the AP integral (Q1322691)

This paper extends to the approximate Perron integral of Burkill [\textit{J. C. Burkill}, Math. Z. 34, 270-278 (1931; Zbl 0002.38604); the reviewer, J. Aust. Mat. Soc., Ser. A 35, 236-253 (1983; Zbl 0533.26005)] the recent limit theorems, known as controlled convergence theorems, that were proved for the Henstock-Kurzweil (= Denjoy-Perron) integral [see \textit{P.- Y. Lee}: ``Lanzhou lectures on Henstock integration'' (1989; Zbl 0699.26004)]. These controlled convergence theorems occur in several forms depending on the kind of uniformity condition applied to the converging sequence of functions. Some of these conditions are extended to the approximate case and the appropriate convergence theorems are then proved.