16.2 Vector Measures

Definition 16.2.1 (Vector Measure). 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:

$\mu(\emptyset) = 0$.
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 16.2.2. 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

\[\sup_{A \in \cm}\norm{\mu(A)}_{E} < \infty\]

Proof. For each $\phi \in E^{*}$, the mapping

\[\mu_{\phi}: \cm \to \complex \quad A \mapsto \dpn{\mu(A), \phi}{E}\]

is a complex measure. For each $A \in \cm$, let

\[x_{A}: E^{*} \to \complex \quad \phi \mapsto \dpn{\mu(A), \phi}{E}\]

then for each $\phi \in E^{*}$,

\[\sup_{A \in \cm}|\dpn{\phi, x_A}{E^{**}}| = \sup_{A \in \cm}|\dpn{\mu(A),\phi}{E}| < \infty\]

By Proposition 9.3.4 and Proposition 9.3.2, $E^{*}$ is a Banach space. The Uniform Boundedness Principle implies that

\[\sup_{A \in \cm}\norm{\mu(A)}_{E}= \sup_{A \in \cm}\norm{x_A}_{E^{**}}< \infty\]

$\square$

Jerry's Digital Garden

16.2 Vector Measures

Direct References

Jerry's Digital Garden

Direct References