Let $X$ be a compact space, $C(X)$ the space of continuous functions on $X$, and $\mathscr{M} = C(X)^*$, the dual of $C(X)$ which we identify with the space of all complex Baire measures. Let $$\mathscr{M}_+(X) = \{ \ell \in \mathscr{M}(X)| \text{ $\ell$ is a positive continuous linear functional}\} \\ \mathscr{M}_{+,1}(X) = \{\ell \in \mathscr{M}_+ ~\big\vert~ \|\ell\| = 1\}.$$
For any $\mu \in \mathscr{M}_{+,1}$ and $g \in X$, let $T_g\mu$ be defined by $T_g \mu(f) = \mu(f_g)$ where $f_g(h) = f(h-g)$.
I would like to show that $T_g$ is continuous in the weak* topology. Let $\{\mu_n\}$ be any sequence of measures in $\mathscr{M}_{+,1}$ such that $\mu_n \rightarrow \mu$ in the weak* topology, which means $$\mu_n(f) \rightarrow \mu(f), \quad \forall f \in C(X).$$ Then for any $f$, $$T_g \mu_n(f) = \mu_n(f_g) \rightarrow \mu(f_g) = T_g \mu(f)$$ so $T_g$ preserves convergence of sequences. However, I do not have any reason to believe that the weak* topology is first countable so I am not sure if this is sufficient or correct.
What is the standard approach to showing that a map between two dual spaces is weak* continuous? I do not see how to prove this straight from the definition of continuity (preimage of open sets being open).