Corollary 15.4.4 (Dominated Convergence Theorem (In Measure)).label Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g, h \in L^{p}(X; \real)$ such that:
- (M)
$\fF \to g$ locally in measure.
- (D)
There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$.
then $\fF \to f$ in $L^{p}(X; E)$. In particular, if $p = 1$, then
\[\lim_{f, \fF}\int f d\mu = \int g d\mu\]
Proof. Since (D) implies (UI) and (T) of the Vitali Convergence Theorem, the result follows from the Vitali Convergence Theorem.$\square$
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