convergence in Lp, convergence for p smaller infinity

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?

Re: convergence in Lp, convergence for p smaller infinity

Try .

Re: convergence in Lp, convergence for p smaller infinity

and

but why doesn't converge

to 0 ?

and

I'm not sure because for me converges to the empty set so why doesn't go to 0 ?

Re: convergence in Lp, convergence for p smaller infinity

Sorry, I meant for the first example. For the second one, take .

Re: convergence in Lp, convergence for p smaller infinity

Hi

Well, but ???

And how can I calculate ?? It's not a Riemann integral...?

Regards

Re: convergence in Lp, convergence for p smaller infinity

Maybe is better for the first question.

For the computation of the integral, use the substitution .

Re: convergence in Lp, convergence for p smaller infinity

Hi

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 ???

Regards

Re: convergence in Lp, convergence for p smaller infinity

I've edited for the first; for the second is just an improper integral (I "cheated" a little as the function is not in ).

Re: convergence in Lp, convergence for p smaller infinity

Hi

Thank you.

1st:

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?

2nd:

(*)

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?

Regards (Nod)

Re: convergence in Lp, convergence for p smaller infinity

Yes asb is convergent. But I cheated a little as I took an unbounded function.

Re: convergence in Lp, convergence for p smaller infinity

Hi

1st: You just didn't answer my question. What about my arguments concerning the first part?

2nd: Why isn't for n to infinity?

I don't understand ??

Re: convergence in Lp, convergence for p smaller infinity

You are right for the first question (I forget answering you before, sorry...).

For the second one, is not essentially bounded (otherwise on would be so).

Re: convergence in Lp, convergence for p smaller infinity

Hi

But your "cheating" isn't a problem at all. The function has to be in L_p for all p only... it can be unbounded?

Re: convergence in Lp, convergence for p smaller infinity

Yes, as if , we should have for each integer , which is not possible.

Re: convergence in Lp, convergence for p smaller infinity

So the reason here is formally said for the 2nd part:

but

and

---doesn't work