What does it mean for a sequence $f_n$ to converge to some function, say, $f$ in the $L^1$ norm?
Is it enough to show that $\int|f_n -f| \to 0$ or must one show as well that $f\in L^1$?
I am getting confused because I've encountered questions which asked to show that $f\in L^1$ and $\lim_{n\to \infty} \int |f_n-f| =0$. Does the latter imply the former?