Fundamental Theorem of Calculus for Lebesgue Integrable functions

Question

I am trying to prove the following statement:
If $f$ is Lebesgue integrable on $[a,b]$ and if $F(x)=\int_{a}^x f(t)dt$
then $𝐹′=𝑓(𝑥)$ 𝑎.𝑒. on $[𝑎,𝑏]$.
If $f$ is Riemann integrable it can be shown using the theorem that -"If $f$ is Riemann integrable, it is discontinuous in the set of measure of zero."
How to prove this statement for non-Riemann integrable functions?

Answer 1

This theorem is definitely nontrivial and is at the basis of the theory that was published originally in Lebesgue's dissertation Intégrale, longueur, aire ("Integral, length, area") at the University of Nancy during 1902.

The main idea is to rephrase this statement in terms of measures and, with the help of measure-theoretic machinery, to solve the corresponding "paraphrased" problem with the help of Radon-Nikodym theorem and the notion of absolute continuity of measures. Finally, one goes back to the functional setting and gets the result.

A formal proof needs a bit of work, so I suggest that you consult books like Modern Real Analysis, by Ziemer or Real Analysis: Modern Techniques and Their Applications, by Folland.