TITLE: If $f_n \in L^1$, will the limit function $f$ also be in $L^1$ in monotone convergence theorem?
QUESTION [5 upvotes]: Monotone convergence theorem doesn't require the sequence of functions $f_n$'s to be $L^1$. When $f_n\in L^1$, will its pointwise limit function $f$ also be in $L^1$? Thanks!

REPLY [5 votes]: Not necessarily.
Define $f_n$ as follows.
On $[-n,n]$, set $f_n(x):=1$ and $f_n(x):=0$ elsewhere.
Each $f_n$ is in $L^1$.
But the pointwise limit, which is equal to the constant function $f(x)=1$ on $\mathbb{R}$, is not in $L^1$.

REPLY [2 votes]: Not quite, since the monotone convergence theorem includes the case where $\lim_{n \to \infty} \int f_n \ d\mu = \infty$.  But if the limit is finite, $f \in L^1$.