Let $(X,S,\mu)$ be a measure space, and let $f,f_1,f_2,\dots:X\to [0,+\infty]$ be $\mu$-integrable such that $\lim\limits_{n\to\infty}f_n=f$ almost everywhere. Show that: $\lim\limits_{n\to\infty}\displaystyle\int|f_n-f|=0 \Leftrightarrow \lim\limits_{n\to\infty}\displaystyle\int|f_n|=\displaystyle\int|f|$ in which case $\lim\limits_{n\to\infty}\displaystyle\int f_n=\displaystyle\int f$.
"$\Rightarrow$" is not hard to show. But how to show "$\Leftarrow$"?