A Uniform Convergence Theorem for Approximate Henstock-Stieltjes Integral

Abstract

In this paper, we introduce, for each approximate distributiontilt of [a,b], the approximate Henstock-Stieltjes integralwith value in Banach spaces. The Henstock integral is a specialcase of this where tilt={(tau,[a,b]): tauin[a,b]}. Thisnew concept generalizes Henstock integral and abstractPerron-Stieltjes integral. We establish a uniform convergencetheorem for approximate Henstock-Stieltjes integral, which is animprovement of the uniform convergence theorem forPerron-Stieltjes integral by Schwabik [3].