Let $(X,\tau,\mu)$ a measure space and $(f_n)_n$ monotone sequence of functions in $L^1$ such that $f_n\to f$ pointwise.
Show that $\sup_{n\in\mathbb{N}} |\int f_n|<\infty$ then $f\in L^1$
Why $\sup_{n\in\mathbb{N}} |\int f_n|<\infty$ implies $f\in L^1$?
If $\{f_n\}$ increases to $f$ then we can apply Fatou's Lemma to $\{f_n-f_1\}$ to get $\int (f-f_1) \leq \lim \inf \int (f_n -f_1) \leq \sup_n |\int f_n |-\int f_1 <\infty$ . This shows $f-f_1 \in L^{1}$, so $f =f-f_1 +f_1 \in L^{1}$. In the case of a decreasing sequence simply replace $f_n$ by $-f_n$ and $f$ by $-f$.