Proof of Theorem on Differentiability and Lebesque Integral

[Nusc]

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.

 

[morphism]

He proves that in his Real & Complex Analysis book. It's Theorem 8.21 on page 169 of the first edition.

 

[Nusc]

Thank you dearly.

 

[Nusc]

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?

 

[morphism]

Have you tried to prove them yourself? They easily follow from their real-valued analogues.

 

[Nusc]

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?

 

[Nusc]

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>

 

[morphism]

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?

 

[Nusc]

It was from baby rudin on page 324. I can only find the proof for the converse in Real and Complex Analysis.