Essential Supremum Problem: Measurable Positive Functions

[johnson123]

Problem: Show an example of a sequence of measurable positive functions on (0,1) so that
$$\left\|\underline{lim} f_{n}\right\| < \underline{lim}\left\|f_{n}\right\| for n\rightarrow\infty$$

My work: I think its just the indicator function $$I_{[n,n+1]}$$

Since $$\left\|\underline{lim} I_{[n,n+1]}\right\|= 0 < \underline{lim}\left\|I_{[n,n+1]}\right\| =1$$

For some reason I do not feel to confident in my answer, so any comments are welcome.

 

[johnson123]

correction

Problem: Show an example of a sequence of measurable positive functions on (0,1) so that
$$\left\|\underline{lim} f_{n}\right\|_{\infty} < \underline{lim}\left\|f_{n}\right\|_{\infty} for n\rightarrow\infty$$

My work: I think its just the indicator function $$I_{[n,n+1]}$$

Since $$\left\|\underline{lim} I_{[n,n+1]}\right\|_{\infty}= 0 < \underline{lim}\left\|I_{[n,n+1]}\right\|_{\infty} =1$$

For some reason I do not feel to confident in my answer, so any comments are welcome.

 

[Dick]

That's pretty good. But the domain of the indicator functions isn't (0,1). Can you build a very similar example using functions defined only on (0,1)?

 

[johnson123]

Thanks for the response Dick.

If $$f_{n}=I_{(\frac{n-1}{n},1)}, then \left\|\underline{lim} I_{(\frac{n-1}{n},1)}\right\|_{\infty}= 0 < \underline{lim}\left\|I_{(\frac{n-1}{n},1)}\right\|_{\infty} =1$$

Please correct me if I am wrong.

 

[Dick]

Sure. That's fine. I was thinking of I_(0,1/n), but you can put stuff on the other side of the interval as well.