Proof of Theorem on Differentiability and Lebesque Integral

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Discussion Overview

The discussion revolves around the proof of a theorem related to differentiability and the Lebesgue integral. Participants are seeking detailed proofs for specific theorems and exploring the relationships between differentiability, Lebesgue integrability, and continuity.

Discussion Character

  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • One participant inquires about a proof for a theorem stating that if F is differentiable on [a,b] and F' is Lebesgue integrable, then F(x) equals the integral of f from a to x.
  • Another participant references a specific theorem from a book, indicating that the proof can be found on page 169 of the first edition of "Real & Complex Analysis."
  • A different participant asks for proofs of several theorems extended to Lebesgue integrals of complex functions, suggesting that they may follow from real-valued analogues.
  • There is a clarification regarding the actual theorem concerning the relationship between f in L and F'(x) being equal to f(x) almost everywhere on [a,b].
  • One participant expresses confusion about the continuity aspect mentioned in the theorem and seeks clarification on what continuity is being referred to.
  • Another participant notes that the theorem is from "Baby Rudin" and mentions difficulty finding the proof for the converse in the referenced book.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the existence of a detailed proof for the theorem in question, and there are multiple competing views regarding the proofs and theorems referenced.

Contextual Notes

There are unresolved questions regarding the continuity aspect of the theorem and the specific references to theorems in various texts. Some assumptions about the relationships between theorems and their analogues remain unexamined.

Nusc
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Hello,


I was wondering where I can find a proof to the following theorem:

If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then

F(x) = int(f,dt,a,x) a<=x<=b.


And the converse.


He gives the theorem are page 324 and a reference in his bibliography. I was wondering where I detailed proof for this theorem.
 
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He proves that in his Real & Complex Analysis book. It's Theorem 8.21 on page 169 of the first edition.
 
Thank you dearly.
 
Hey morphism,

do you know where I can find the proofs for the theorems 11.23 (a), (d), (e), (f), 11.24(b), 11.26, 11.27, 11.29, and 11.32 extended to Lebesgue integrals of complex functions?
 
Have you tried to prove them yourself? They easily follow from their real-valued analogues.
 
He just gives the converse to

If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then

F(x) = int(f,dt,a,x) a<=x<=b.

Where can I find the proof for this?
 
Sorry, the actual theorem is:

If f in L on [a,b] and F(x)=int(f,t,a,x) (a<=x<=b) then F'(x)=f(x) almost everywhere on [a,b].

and he uses strictly eveywhere for continuity, why is this>
 
I'm not sure that I understand what it is you're asking.

Nusc said:
He just gives the converse to

If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then

F(x) = int(f,dt,a,x) a<=x<=b.

Where can I find the proof for this?
Like I said, this is in one of his other books (with your typos corrected!), Real and Complex Analysis.

Nusc said:
Sorry, the actual theorem is:

If f in L on [a,b] and F(x)=int(f,t,a,x) (a<=x<=b) then F'(x)=f(x) almost everywhere on [a,b].

and he uses strictly eveywhere for continuity, why is this>
Where is this from? And continuity of what is being used?
 
It was from baby rudin on page 324. I can only find the proof for the converse in Real and Complex Analysis.
 

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