So I'm studying for an exam, and this is one of the problems stumping me. First my book defines a distribution function as the following $F:\mathbb{R} \to \mathbb{R}$ is a distribution function if $F$ is monotone nondecreasing, $F$ is right continuous, $F$ is bounded, and $\lim_{x\rightarrow -\infty} F(x) = 0$. The problem is then as follows: Let $F$ be a distribution function, and suppose that $F$ is a continuously differentiable. Let $\mu$ be the associated measure. Show that $f=F'$ is integrable and that $\mu(E) = \int_E f d\mu \forall E \in \mathcal{B}$ where $\mathcal{B}$ are the Borel sets.
So this was my attempt:
Since $F$ is continuously differentiable, both $F,F'$ are continuous, and so $F'$ is bounded. Since $f=F'$ we have that $f$ is bounded, and thus $\int f d\mu <\infty$. So $f$ is integrable. I suppose I'm not sure how to relate $F$ back to $\mu$ or to show that the formula I am asked to prove is valid.