  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.
