Partial Integration for measures

Question

I have the following formula in mind, $\mu$ a measure on $\mathbb{R}$. Any sigma-finite measure on $\mathbb{R}$ can be decomposed into a absolut continuous part, a "point measure" and a singular continuous part. For this measure $\mu$ the first two parts are 0. Now, is the following formular true: $$ \int_a^b f(t)d\mu (t)=f(t)\mu(]-\infty,t])\bigg |_a^b-\int_a^b f'(t)\mu(]-\infty,t])dt $$ ? I know the formula holds for absolut coninuous measures and for point measure (under the assumption that $\mu (a)=\mu(b)=0$. But does it hold for all measures?

Answer 1

Found the answer myself: $$ \int_a^b f(t)\mu (dt)=f(a)\mu([a,b])+\int_a^b\int_a^b \chi _{[a,t]}(s)f'(s)d\mu(t)ds= \text{Claim} $$