Rudin Reference

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  • #1
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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.
 

Answers and Replies

  • #2
morphism
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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.
 
  • #3
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Thank you dearly.
 
  • #4
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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?
 
  • #5
morphism
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Have you tried to prove them yourself? They easily follow from their real-valued analogues.
 
  • #6
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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?
 
  • #7
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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>
 
  • #8
morphism
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I'm not sure that I understand what it is you're asking.

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.

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?
 
  • #9
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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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