Linear continuous functionals on the space $(BV)$ with weak topologies

In two recent papers, G. L. Krabbe [6], [7] has obtained in a general setting a theorem which for the special case when the functions involved are complex valued, yields a representation for the linear continuous functional on the space of functions of bounded variation (BV) on a finite interval with the weak topology of pointwise convergence. Some of the derivation uses the algebraic properties of the space (BV). We derive here the special case when the functions are complex valued, since it can be obtained by simple analysis methods and incidentally brings out relations between modifications of the Stieltjes integral definition which allow for the existence of the integral when the two functions involved have simultaneous discontinuities. The derivation also suggest the determination of a corresponding representation when the weak topology on (BV) is strengthened to uniform convergence.