The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.

Here I come up with a question which has similar assumption as Riesz representation but different in some way, and I am wondering that could I make some conclusion similar to Riesz representation.

Suppose $\mu\in\mathcal M(I)$ where $I\subset \mathbb R^1$ is open interval and $\mathcal M(I)$ denotes all fnite Rodan measures. Assume $$ \sup\left\{\int_I \phi'\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(I)}\leq1,\,\|\phi'\|_{L^\infty}\leq 1\right\}<\infty $$ Then I am wondering that can I have some similar result compare to Riesz representation? Since in the integration is $\phi'$ but not $\phi$ and the constraint is on $W^{1,\infty}$ norm and it is much stronger then the $L^\infty$ norm in usual Riesz representation, I am not expecting I will have same result as in Riesz. But would it be possible to conclude something similar to integration by parts formula? i.e., to have another Radon measure $\nu$ such that $$ \int_I\phi'\,d\mu = -\int_I\phi\,d\nu $$ Any help is really welcome!