Sufficient Conditions for Convergence of Sequences of Henstock-Kurzweil Integrable Functions

Abstract

The main aim of this paper is to present our approach of obtaining sufficient conditions for convergence of sequences of Henstock-Kurzweil integrable functions. Our approach involves the use of the concept of multiplier functions, where we define a class Φ of multipliers for the Henstock-Kurzweil integral. We consider a sequence (fn) of Henstock-Kurzweil integrable functions on a non-degenerate interval [a, b] and we assume that (fn) converges point wise to a function f. Then we show that f is Henstock-Kurzweil integrable and its integral is equal to the limit of the sequence (∫abfn) if there exists φ∈Φ such that the defined functionals of the type F (φ, fn) satisfy the imposed conditions. Beside the fact that the results regarding the convergence under the integral sign are always of great importance, the method introduced here can be imitated and used to obtain other results on the related areas.