Let X=[0,1] , A= . (B is the Borel- algebra), the lebesgue measure. In explain why these statements are true or false:
1) if with and then
2) if and then
Well I honestly do not have much ideas how to solve this. It's clear 1) isn't true because 2) wouldn't make sense if 1) was true. But a formal explanation? A prove? I don't know.
All I tried was finding functions that are for example in L_1 but not in L_2 . But I couldn't find a contradiction. Also there exist Lebesgue-integrable functions that are not bounded...??
Could please someone give me a hint?
What does mean? I've never seen this before. The square root of an interval???
The substitution didn't help here either. I don't know what to do because this is not a Riemann integral ????
so in the end this integral is simply 0 ???
So isn't a zero set for all n so it isn't one when n goes to infinity as well?
then for all and this is not bounded.
This is the correct reason for that yes?
But what is ? Is it smaller than infinity?
But here when n goes to infinity (*) goes to zero and so does the pth root of (*) or what did I do wrong here?