Maximally strong fundamental theorem

[jostpuur]

This is still not clear to me. Here's the conjecture:


Assume that $f:[a,b]\to\mathbb{R}$ is such function that it is differentiable at all points of its domain, and that

$$\int\limits_{[a,b]}|f'(x)|dm(x) < \infty$$

holds, where the integral is the ordinary Lebesgue integral. Then also

$$\int\limits_{[a,b]}f'(x)dm(x) = f(b)-f(a)$$

holds.


True or not? I don't know a proof, and I don't know a counter example.

 

[Office_Shredder]

Can you explain the difference between this and the ordinary fundamental theorem of calculus, because I can't see it.

 

[UltrafastPED]

I think its true, but you can search for a proof:
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf

see theorem 8.40, though it uses different notation

 

[jostpuur]

hmm... I just realized that actually I did have a proof for this. I had only forgotten about it and lost it, hehe (I'm still not sure if the proof is right though...)

The difference is in the assumptions.

In the ordinary theorem it is assumed that the derivative is Riemann integrable.

The theorem can be made slightly stronger pretty easily by only assuming that the derivative is bounded.

But when you only assume that the derivative is Lebesgue integrable, stuff gets serious.

 

[jostpuur]

I think its true, but you can search for a proof:
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf

see theorem 8.40, though it uses different notation

This theorem seems to assume that the derivative is continuous.

My own proof was in this thread from the summer: https://www.physicsforums.com/showthread.php?t=696481&page=2 I had forgotten about this, but I checked my old threads now... There was one intermediate result whose proof was left uncertain.