Assume that $f\colon \Bbb R \rightarrow\Bbb R$ is left-continuous nondecreasing and let $\mu$ be a Borel measure in $\Bbb R$ such that $\mu([a,b))=f(b)-f(a)$ for $a, $a,b \in\Bbb R$.
I would like to prove that $\int \phi \,d\mu=-\int\phi'(x)f(x)\,dx$ for each $\phi\colon \Bbb R\rightarrow\Bbb R$ smooth with compact support.
How to show that LHS and RHS in the above equality are equal $\int \int_{ \{(x,y)\in\Bbb R\times\Bbb R:x