For a compact space $X$ I want to show that the space of finite signed measures on the Borel-$\sigma$-algebra on $X$, equipped with the weak*-topology, i.e. the corsest topology such that
$\mu\mapsto\int f~\mathrm{d}\mu~$ is continuous for all $f\in C_b(X)$ is a topological vector space.
Therefore I think I need to show that the maps $(\mu,\nu)\mapsto\mu+\nu$ and $(c,\mu)\mapsto c\mu$ are continuous.
I tried to do it with the formal definition of continuity in topological spaces, i.e. I wanted to show that $\{(\mu,\nu):\mu+\nu\in U\}$ and $\{(c,\mu):c\mu\in U\}$ are open in the topology for open $U$ but I have no idea how to do that.
Can someone give me a reference or a hint on how to do that?
Best Answer
Perhaps you should do it by nets:
Assume that $(\mu_{\alpha},\nu_{\alpha})\rightarrow(\mu,\nu)$ weak$^{\ast}$, then $\displaystyle\int fd(\mu_{\alpha}+\nu_{\alpha})=\int fd\mu_{\alpha}+\int fd\nu_{\alpha}\rightarrow\int fd\mu+\int fd\nu=\int fd(\mu+\nu)$, so $\mu_{\alpha}+\nu_{\alpha}\rightarrow\mu+\nu$ weak$^{\ast}$. For the scalar multiplication is similar.