If $\{f_n\}$ is a set of increasing functions defined on $[a,b] \subset \mathbb R$ and $\sum_{n=1}^{+\infty} f_n$ converges to $F(x)$, will $\sum_{n=1}^{+\infty} f_n$ be Lebesgue measurable?
I got this question when I was reading Fubini's Differentiation Theorem. For finite sum, that's correct. But I'm not sure for infinite sum.
Besides, I've no idea of a more general case that is removal of limitation of "increasing" functions, in other words, a set of measurable functions, will this question have the same answer?
Increasing functions are measurable. Pointwise limits of measurable functions are measurable. Since the sum converges at each point $x$ to a point on the extended real line, try looking at the partial sums, which are measurable.