Theorem 20.2.2 (Dominated Convergence Theorem). Let $(X, \cm, \mu)$ be a measure spacs, $E$ be a Banach space over $K \in \RC$, $\seq{f_n}\subset L^{1}(X; E)$, and $f \in L^{1}(X; E)$. If
$f_{n} \to f$ strongly pointwise.
There exists $g \in L^{1}(X) \cap L^{+}(X)$ such that $\norm{f_n}_{E} \le g$ for all $n \in \natp$.
then $\int f d\mu = \limv{n}\int f_{n} d\mu$.
Proof. By the classical Dominated Convergence Theorem, $f_{n} \to f$ in $L^{1}(X; E)$. Since $h \mapsto \int h d\mu$ is a bounded linear operator, $\int f d\mu = \limv{n}\int f_{n} d\mu$.$\square$