On integrals with integrators in BVp

Abstract

n 1936, L. C. Young proved that the Riemann-Stieltjes integral$\int^b_a$ $f$ $dg$ exists, if $f\in BV_p,\,g\in BV_q, \frac{1}{p}+\frac{1}{q}>1$ and $f,g$ do not have common discontinuous points. In this note, using Henstock's approach, we prove that $\int^b_a$ $f$ $dg$ still exists without assuming the condition on discontinuous points. Some convergence theorems are also proved.