Proposition 21.7.3.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:

1. (P)

   For each $x \in X$, $\bracs{x}\in \cm$, and the delta mass $\delta_{x}$ is in $\mathscr{M}$.

Then, for any bounded measurable functions $\bracsn{f_n: X \to E^*|n \in \natp}$ and $f: X \to E^{*}$, the following are equivalent:

1. (1)

   For each $\mu \in \mathscr{M}$, $\limv{n}\int f_{n} d\mu = \int f d\mu$.

2. (2)

   For each $x \in X$, $\limv{n}f_{n}(x) = f(x)$, and $\sup_{n \in \natp}\norm{f_n}_{u} < \infty$.