- #1
- 10,877
- 422
Define ##\rho(f)=\int |f|\mathrm d\mu## for all integrable ##f:X\to\mathbb C##. This ##\rho## is a seminorm, not a norm. Does ##\rho(f_n-f)\to 0## imply ##f_n\to f## a.e.?
I kind of think that it should, because in the case of real-valued functions, ##\int|f_n-f|\mathrm d\mu## is the area between the graphs, and I don't see how it can go to zero without ##f_n\to f## a.e. I have a feeling that this should be easy, but I don't see how to prove it. Going to bed now. Maybe I'll see it immediately in the morning.
I kind of think that it should, because in the case of real-valued functions, ##\int|f_n-f|\mathrm d\mu## is the area between the graphs, and I don't see how it can go to zero without ##f_n\to f## a.e. I have a feeling that this should be easy, but I don't see how to prove it. Going to bed now. Maybe I'll see it immediately in the morning.