The question "If {$F_j$} is a sequence of nonnegative increasing functions on $[a,b]$ such that $F(x)= \sum_1^\infty F_j(x) < \infty$" for all $x \in [a,b]$, then $F\prime(x)=\sum_1^\infty F\prime_j(x)$ for a.e. $x \in [a,b]$. (It suffices to assume that $F_j \in NBV$. Consider the measures $\mu_{F_j}$.)"
By the theorem 3.23 page 101 in the same book it makes sense to assume that $F_j \in NBV$ for all $j$.
I have very basic questions actually: first of all, how is derivative of F expressed? Are we allowed to write it down as: $F\prime = lim_{r\rightarrow 0} \frac{\mu_F(E_r)}{m(E_r)}$ where $\mu_F$ is the Borel measure , $\mu((a,b])=F(b)-F(a)$, $m$ is the Lebesgue measure, and $E_r = (x,x+h]$.
After that I consider writing $\mu_F$ in terms of $\mu_{F_j}$'s and obtain the equality, but it seems very wrong and there is not any use of the fact that $F_j$'s are nonnegative.
So could anyone please give me some hints and clarifications?
Thank you.