I've got the following two exercises to solve:
Consider the measure space where B denotes the Borel-sigma-algebra and the Lebesgue-measure.
Let and continuous. Show that
is differentiable and that
If is almost everywhere continuous only is then
almost everywhere as well?
Why? Prove your answer!
According to the theorem of Radon-Nikodym 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 one-point set... cause then it was all zero.
I'd say that this is right because where
is a Lebesgue-zero set but how to show?
Could please someone help me to solve this exercise? Any hint would be appreciated.
Thank you very much.
A kind of "prove" or argument hmmm
assuming g is continuous at and is contained in a non-zero-set 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?
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?