Proposition 23.7.3.label Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mathscr{M}\subset M(X, \cm; E)$ be a closed subspace such that:
- (P)
For each $x \in X$, $\bracs{x}\in \cm$, and the delta mass $\delta_{x}$ is in $\mathscr{M}$.
then for any sequence $\seq{f_n: X \to E^*}$ of bounded strongly measurable functions, the following are equivalent:
- (1)
For each $\mu \in \mathscr{M}$, $\limv{n}\int f_{n} d\mu$ exists.
- (2)
There exists a bounded strongly measurable function $f: X \to E^{*}$ such that $f_{n} \to f$ pointwise and $\sup_{n \in \natp}\norm{f_n}_{u} < \infty$.
Proof. (1) $\Rightarrow$ (2): By (P), for each $x \in X$, $\limv{n}f_{n}(x)$ exists. By the Uniform Boundedness Principle,
(2) $\Rightarrow$ (1): By the Dominated Convergence Theorem.$\square$
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