Lebesgue decomposition of increasing function

Question

Suppose $F$ is right-continuous, and increasing. By lebesgue decomposition, $F=F_A+F_C+F_J$ where $F_A$ is absolutely continuous, $F_C$ is continuous with $F'_C=0$ a.e., and $F_J$ is pure jump function. Let $\mu=\mu_A+\mu_C+\mu_J$ be the Borel measures associated to each function.

I want to prove that $ \mu_A(E) = \int_E F'(x)dx$ for lebesgue measurable set $E$. However, I am unclear how can I get $\int_E F'(x)dx$. Is $F'= F'_A+F'_C+F'_J$?

Answer 1

$F'=F_A'+F_C'+F_J'$ a.e. is true and the second and third terms are both $0$ a.e. So $F'=F_A'$ a.e. From this it follows that $\mu_A(E)=\int_E F_A'=\int_A F'$.