Fatou's lemma and the monotone convergence theorem

Let $\{f_n\}$ be a sequence of measurable functions $f_n:X\to[0,\infty]$. I: $\liminf\int f_n\,d\mu\ge\int\liminf f_n\,d\mu$. II: If $f_n(x)\uparrow$ for each $x$, then $\int\lim f_n\,d\mu=\lim\int f_n\,d\mu$.