How to prove or disprove$$\lim_{n\to\infty}\|f_n-f\|=0\;\Rightarrow \;\lim_{n\to\infty}f_n(x)=f(x)\; a.e.?$$ Any hint is appreciated.
Asked
Active
Viewed 3,556 times
2
-
2This is false. But you can extract a subsequence which converges a.e. See the duplicate. – Julien Apr 15 '13 at 02:43
-
Thank you very much, @julien. I am not sure if I should delete the post. – Sam Apr 15 '13 at 02:57
-
I don't know about that, actually. Good question! – Julien Apr 15 '13 at 02:58
1 Answers
3
This is false. For a counterexample, consider the functions $\chi_{[0,1]}$, $\chi_{[0,1/2]}$, $\chi_{[1/2,1]}$, $\chi_{[0,1/4]}$, $\chi_{[1/4,1/2]}$, $\chi_{[1/2,3/4]}$, and so on.
However, you can always extract a subsequence which does converge a.e.
Potato
- 40,171