Theorem 20.2.4 (Dominated Convergence Theorem). Let $(X, \cm, \mu)$ be a measure space, $E, F$ be normed spaces over $K \in \RC$, $G$ be a Banach space over $K$,
\[\lambda: E \times F \to G \quad (x, y) \mapsto xy\]
be a bounded bilinear map, $\mu: \cm \to F$ be a vector measure, $\seq{f_n}\subset L^{1}(X, |\mu|; E)$, and $f \in L^{1}(X, |\mu|; E)$. 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 $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_{n}, d\mu)$.
Proof. By the classical Dominated Convergence Theorem, $f_{n} \to f$ in $L^{1}(X, |\mu|; E)$. Since $h \mapsto \int \lambda(f , d\mu)$ is a bounded linear operator, $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_{n}, d\mu)$.$\square$