Integration by Parts for a Complex Measure

Question

Let $f$ be a continuously differentiable function defined on $\mathbb{R}$, $\mu$ a complex (Radon) measure on $\mathbb{R}$, and $F(x)=\mu((-\infty,x])$. Why is it true that $$\int f d\mu=\int f'(x)F(x)dx$$ Folland suggests that one should use the integration by part formula (Theorem 3.36 on his Real Analysis), but I did not see how it works.