In this 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.