This is based on an approach to a homework problem from last year that I discarded but would like to fill in the details of now. Unfortunately I can't see the necessary inequality.
Suppose we are given a function $g$ of bounded variation on $[a,b]$. I would like to construct the Lebesgue-Stieltjes integral with respect to $g$ by defining a bounded linear functional on $C^1([a,b])$ by \varphi \mapsto -\int_{\mathbb{R}} g(x) \varphi^{'}(x) dx then extending it to $C([a,b])$ by density and then applying the Riesz Representation Theorem. To do this, however, I need to show that |\int g(x) \varphi^{'}(x) dx|\leq C||\varphi||_\infty (notice that there is no derivative here--this is the condition to make it a bounded functional on $C^0([a,b])$) which is where the trouble comes in since I am not sure how to get such a bound without integrating by parts (which we can't do since $g$ need not be absolutely continuous).
I'm sure there is a nice way to handle this but I just don't see it.
Thanks for the help.