Interchanging sum and differentiation, almost everywhere Let $\{F_i\}$ be a sequence of nonnegative increasing real functions on $[a,b]$ with $a<b$ such that $F(x):=\displaystyle\sum_{i=1}^\infty F_i(x)<\infty$ for all $x\in [a,b],$ then show $F'(x)=\displaystyle\sum_{i=1}^\infty F'_i(x)$ a.e. on $[a,b]$.
At first note that we may assume each $F_i$ is right continuous and can consider the corresponding measure $\mu_{F_i}([a,x])=F_i(x)$. Then.. what is the next step? help me.

Consider the derivarive of $\mu_{F_i}$, $\mu_{F_i}[a,x] = \int_a^x F'_i(s)\, ds$.
Now note that $F'_i \geq 0$.
Compute define $G_N(x):=\left(\sum_{i=1}^N F_i(x)\right)$ 
$$G_N'(x) = \sum_{i=1}^N F'_i(x) = g_N(x) $$
Note that $g_N(x) \uparrow g_\infty(x)$ as $N \to \infty$. 
This implies that $F'(x)= g_\infty(x)= \sum_{i=1}^\infty F'_i(x)$
Since $G_N(x)= = G_N(a) + \int_0^x g_N(s)\, ds $ take $N\to \infty$  and apply monotone convergence theorem to get
$$F(x) = F(a) + \int_a^x g_\infty(s)\, ds $$
Therefore $F'(x) = g_\infty (x)= \sum_{i=1}^\infty F'_i(x)$
