- #1
johnson123
- 17
- 0
Problem: Show an example of a sequence of measurable positive functions on (0,1) so that
[tex]\left\|\underline{lim} f_{n}\right\| < \underline{lim}\left\|f_{n}\right\| for n\rightarrow\infty[/tex]
My work: I think its just the indicator function [tex]I_{[n,n+1]}[/tex]
Since [tex]\left\|\underline{lim} I_{[n,n+1]}\right\|= 0 < \underline{lim}\left\|I_{[n,n+1]}\right\| =1 [/tex]
For some reason I do not feel to confident in my answer, so any comments are welcome.
[tex]\left\|\underline{lim} f_{n}\right\| < \underline{lim}\left\|f_{n}\right\| for n\rightarrow\infty[/tex]
My work: I think its just the indicator function [tex]I_{[n,n+1]}[/tex]
Since [tex]\left\|\underline{lim} I_{[n,n+1]}\right\|= 0 < \underline{lim}\left\|I_{[n,n+1]}\right\| =1 [/tex]
For some reason I do not feel to confident in my answer, so any comments are welcome.