Dominated Convergence Theorem: Let $f_n$ be measurable and assume that $f_n$ converges a.e. to $f$. If $|f_n(x)|\leq g(x)$ for some integrable $g$ it follows that $f_n$ converges in $L^1$ to $f$, in particular $\int f_n\to\int f$. Hence, we do not only have the convergence of the integrals but even $L^1$-convergence.
Now my question: Does the monotone convergence theorem also implies $L^1$ convergence or just $\int f_n\to\int f$?