lebesgue measure and differentiability
Hi there
I've got the following two exercises to solve:

Consider the measure space where B denotes the Borelsigmaalgebra and the Lebesguemeasure.
(1)
Let and continuous. Show that
is differentiable and that
(2)
If is almost everywhere continuous only is then
almost everywhere as well?
Why? Prove your answer!

(1)
According to the theorem of RadonNikodym for then
Obviously the derivative of this is g but how can one show this in a proper way?
????
I also tried it this way:
but what then is x and 0? Definitely not a onepoint set... cause then it was all zero.
(2)
I'd say that this is right because where
is a Lebesguezero set but how to show?
Could please someone help me to solve this exercise? Any hint would be appreciated.
Regards
Huberscher
1 Attachment(s)
Re: lebesgue measure and differentiability
I think you're trying to make an easy problem hard. Here's a solution:
Attachment 26576
Re: lebesgue measure and differentiability
Thank you very much.
Concerning 2)
A kind of "prove" or argument hmmm
assuming g is continuous at and is contained in a nonzeroset of then this proof works also, ok.
But when x satisfies and x is on the left of ...doesn't that mean that at x the function g is continuous from the right since the is arbitraty ? That's right yeah?
Re: lebesgue measure and differentiability
Hi there
Well I understand the proof but I still don't understand why this is also valid for a.e. continuity. Can someone please explain me that?
Cause then I have
is 0 because the set is a zero set but then I still have this which goes to infinity when h goes to 0.
Is there anyone who can answer me that please?
Regards