TITLE: Meaning of convergence in $L^1$ norm
QUESTION [13 upvotes]: 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?

REPLY [4 votes]: The answer of Ragib Zaman is correct. However, I think that a sentence like "$\{f_n\}_n$ converges to $f$ in $L^1$" means:

$f_n \in L^1$ for every $n \in \mathbb{N}$;
$\lim_{n \to +\infty} \int |f_n-f| =0$.

This is a reasonable approach, since, in any (say) metric space $X$, $x_n \to x$ is rather immaterial if $x_n \notin X$. The case of a sequence $u_n = f_n -f \in L^1$ with $f_n \notin L^1$ is really a trap for homework :-)