Let f belong to $C[a,b]$. Show that there is a function g that is of bounded variation on [a,b] for which $\int_a^bfdg=||f||_{max}$ and TV(f)=1.
This problem appears on page 162 of Royden's Real analysis. I have to say that I have no idea how to solve this problem.
You know that there is a point $t_0 \in [a,b]$ where $|f(t_0)| = \|f\|_{\max}$. If $t_0 \in (a,b]$, let $g=\chi_{[t_0,1]}$. Then $\int_{a}^{b}f(t)dg(t)=f(t_0)$. Now multiply $g$ by a unimodular scalar $e^{i\theta}$ in order to arrange $\int_{a}^{b}f(t)d(e^{i\theta}g)(t)=|f(t_0)|=\|f\|_{\max}$. Clearly $V_{a}^{b}(e^{i\theta}g)=1$. A minor modification works if $t_0=a$.