Definition 17.3.3 (Space of Finite Radon Measures). Let $X$ be a LCH space and $E$ be a normed vector space over $K \in \RC$, then $M_{R}(X; E)$ is the space of finite Radon measures on $X$, which forms a vector space over $K$.
Proof. Let $\mu, \nu \in M_{R}(X; E)$, then for any $A \in \cb_{X}$, $|\mu + \nu|(A) \le |\mu|(A) + |\nu|(A)$. Let $\eps > 0$, then by outer regularity and Proposition 17.1.4, there exists $K \subset A$ compact and $U \in \cn^{o}(A)$ such that $(|\mu| + |\nu|)(A \setminus K), (|\mu| + |\nu|)(U \setminus A) < \eps$. Therefore $|\mu + \nu|$ is regular on all Borel sets, and hence Radon.$\square$