A convergence theorem for $ap-$Henstock-Kurzweil integral and its relation to topology

Abstract: In this {\color{red}{paper}} we discuss about the $ap-$Henstock-Kurzweil integrable functions on a topological vector spaces. Basic results of $ap-$Henstock-Kurzweil integrable functions are discussed here. We discuss the equivalence of the $ap-$Henstock-Kurzweil integral on a topological vector spaces and the vector valued $ap-$Henstock-Kurzweil integral. Finally, several convergence theorems are studied.