If $F$ is right continuous and monotonically increasing, then $\int^{a}_b F'(x)dx\leq F(b)-F(a)$

Question

How do I show that if $F$ is right continuous monotonically increasing, then $\int^{a}_b F'(x)dx\leq F(b)-F(a)$?

First, I note that $F$ is differentiable almost everywhere so the integral makes sense. The above claim is heuristically obvious because, at any discontinuity points, $F(b)-F(a)$ can only increase. Additionally, I need to use the fundamental theorem of calculus (which I can only use in the continuous function). Then I feel like in the proof of the claim, I need to use the techniques that are used in the fundamental theorem of calculus. How do I make it rigorous?

Answer 1

$\int_a^{b} \frac {F(x+\frac 1n)-F(x)} {1/n}dx=n\int_b^{b+\frac 1n} F(x)dx-n\int_a^{a+\frac 1n} F(x)dx \to F(b)-F(a)$ by right continuity. Now apply Fatou's Lemma.