The approximate variational integral (Q2520032)

The concept of the generalized approximately continuous Perron integral (GAP) was introduced by \textit{D. K. Ganguly} and \textit{R. Mukherjee} [Math. Slovaca 58, No. 1, 31--42 (2008; Zbl 1164.26010)]. The variational integral is a kind of non-absolute integral originally defined by \textit{R. Henstock} [``Linear analysis'' (1967; Zbl 0172.39001)]. In this paper, the authors gave a characterization of the variational integral by the GAP-integral. They proved that a function is approximately variationally integrable if and only if it is GAP-integrable. They also gave some significant convergence theorems of the GAP-integral using the approximate variational integral.