20.2 Vector Measures
Definition 20.2.1 (Vector Measure).label Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$, then $\mu$ is a vector measure if:
- (M1)
$\mu(\emptyset) = 0$.
- (M2)
For any $\seq{A_n}\subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}A_n}= \sum_{n \in \natp}\mu(A_{n})$ where the sum converges absolutely.
Proposition 20.2.2.label Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$ be a vector measure, then
Proof. For each $\phi \in E^{*}$, the mapping
is a complex measure. For each $A \in \cm$, let
then for each $\phi \in E^{*}$,
By Proposition 11.4.5 and Proposition 11.4.3, $E^{*}$ is a Banach space. The Uniform Boundedness Principle implies that
$\square$