Maximally strong fundamental theorem

In summary, the conjecture states that if a function is differentiable at all points of its domain and the integral of its absolute derivative is finite, then the integral of the derivative is equal to the function's difference at the endpoints. This is known as the fundamental theorem of calculus for Lebesgue integrals. Although there is some uncertainty about the proof, it is generally believed to be true and can be found in resources such as theorem 8.40 in the linked lecture notes. The key difference between this theorem and the ordinary fundamental theorem of calculus is the assumption that the derivative is Lebesgue integrable, which makes the proof more challenging.
  • #1
jostpuur
2,116
19
This is still not clear to me. Here's the conjecture:


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

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

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

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

holds.


True or not? I don't know a proof, and I don't know a counter example.
 
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  • #2
Can you explain the difference between this and the ordinary fundamental theorem of calculus, because I can't see it.
 
  • #4
hmm... I just realized that actually I did have a proof for this. I had only forgotten about it and lost it, hehe :-p (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.
 
  • #5
UltrafastPED said:
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.
 

Related to Maximally strong fundamental theorem

1. What is the "Maximally strong fundamental theorem"?

The Maximally Strong Fundamental Theorem is a mathematical concept that states that for any given set of axioms, there exists a set of theorems that cannot be proven or disproven within that set of axioms. This means that there are limits to what can be proven using a specific set of rules and assumptions.

2. How does the "Maximally strong fundamental theorem" affect mathematics?

The Maximally Strong Fundamental Theorem has important implications for mathematics, as it shows that there are always going to be unprovable statements within any logical system. This means that there will always be room for further exploration and discovery in mathematics.

3. Is there evidence to support the "Maximally strong fundamental theorem"?

Yes, there is evidence to support the Maximally Strong Fundamental Theorem. One example is Gödel's incompleteness theorems, which mathematically prove the existence of unprovable statements within a given set of axioms.

4. Can the "Maximally strong fundamental theorem" be applied to other fields of study?

While the concept of the Maximally Strong Fundamental Theorem was originally developed in mathematics, it can also be applied to other fields of study such as philosophy and logic. It ultimately relates to the idea of the limits of knowledge and understanding within a given system.

5. How does the "Maximally strong fundamental theorem" impact the pursuit of absolute truth?

The Maximally Strong Fundamental Theorem suggests that there may be limits to what we can know and prove within a logical system, meaning that the pursuit of absolute truth may be limited by the boundaries of our understanding and reasoning. It highlights the importance of ongoing questioning and exploration in the pursuit of knowledge and understanding.

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