Recently Bartle [B], modifying a convergence theorem of Gordon, has established necessary and sufficient conditions for a Henstock integral convergence theorem. We point out that this result is not specific to the Henstock, or indeed to any definition of, an integral. Rather, it is a result about an arbitrary limit process and can thus be formulated in the setting of abstract convergence theory.
"The convergence content of the integral convergence theorem." Real Anal. Exchange 21 (2) 771 - 773, 1995/1996.