Lebesgue Fundamental Theorem of Calculus problems

Question

I have a couple problems related to the Lebesgue FTC that I'm having a hard time solving:

1. If $F:\mathbb R\to \mathbb C\in$NBV, show that there is a null Borel set $N\subset\mathbb R$ s.t. if $[a,b]\cap N=0,$ then $$F(b)-F(a)=\int_a^bF'(t)dt$$.
2. Show that if $F:\mathbb R \to \mathbb R$ is increasing and $-\infty<a<b<\infty$, then $$F(b)-F(a)\geq \int_a^bF'(t)dt$$

For both of the problems, I'm kind of at an impasse and just running around in circles. I would greatly appreciate some help on solving these problems.