Definition 16.5.4 (Space of Finite Measures). Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $M(X, \cm; E)$ be the set of all finite $E$-valued vector measures on $(X, \cm)$. For each $\mu \in M(X, \cm; E)$, let $\norm{\mu}_{\text{var}}= |\mu|(X)$, then:
$M(X, \cm; E)$ equipped with $\norm{\cdot}_{\text{var}}$ is a normed vector space over $K$.
If $E$ is a Banach space, then so is $M(X, \cm; E)$.
Proof. (1): For any $\mu, \nu \in M(X, \cm; E)$ and $\seqf{A_j}\subset \cm$ such that $X = \bigsqcup_{j = 1}^{n} A_{j}$,
As this holds for all choices of $\seqf{A_j}$, $\norm{\mu + \nu}_{\text{var}}\le \norm{\mu}_{\text{var}}+ \norm{\nu}_{\text{var}}$.
(2): Let $\seq{\mu_n}\subset M(X, \cm; E)$ such that $\sum_{n \in \natp}\norm{\mu_n}_{\text{var}}< \infty$. For each $A \in \cm$, since $E$ is complete, let
then for any $\seq{A_k}\subset \cm$ and $A \in \cm$ with $A = \bigsqcup_{k \in \natp}A_{k}$,
by Fubini’s theorem.
$\square$