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

TL;DR

This paper explores the properties and convergence theorems of the $ap$-Henstock-Kurzweil integral within topological vector spaces, establishing equivalences and foundational results for this integration approach.

Contribution

It introduces the $ap$-Henstock-Kurzweil integral in topological vector spaces and proves key convergence theorems and equivalence with vector-valued integrals.

Findings

Established basic properties of $ap$-Henstock-Kurzweil integrable functions.

Proved the equivalence between $ap$-Henstock-Kurzweil integral and vector-valued integrals.

Derived several convergence theorems for the $ap$-Henstock-Kurzweil integral.

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.