On the equivalence of four convergence theorems for the AP-integral (Q1322623)

The author of this paper extends the concepts of P-Cauchy to the AP- integral, as AP-Cauchy. The original concept is due to \textit{R. A. Gordon} [Real. Anal. Exch. 18, No. 1, 261-266 (1993; Zbl 0780.26007)], who proved that \(\{f_ n\}\) is uniformly Henstock integrable iff \(\{F_ n\}\) is generalized P-Cauchy; here \(F_ n\) is an AP-primitive of \(f_ n\), \(n= 1,2,\dots\;\). The author extends this result to the AP-integral using his generalized AP-Cauchy concept and recent results of Liao and Chew on controlled convergence for the AP-integral.