We consider a locally compact Hausdorff space $X$ and the Banach space $C_0(X)$ of continuous functions on $X$ taking values at $\mathbb K = \mathbb R$ or $\mathbb C$, equipped with the supremum norm.

I'm studying this paper: http://www.siue.edu/MATH/kj_papers/Into.pdf. At page 375, in the proof of Lemma 2, it says the following:

"If $A$ is a closed subspace of $C_0(X)$, then any functional $F\in A^\ast$ can be extended, with the same norm, to a regular Borel measure $\mu_F$ on $X$."

Until this point, there is no big deal. We use Hahn-Banach to extend $F$ to $\tilde F\in(C_0(X))^*$ with the same norm then use Riesz Representation Theorem to get a regular Borel measure $\mu_F$ at $X$ such that $$ \tilde F(f)=\int f d\mu_F,\ \forall f \in C_0(X). $$ Continuing:

"On the other hand, $A$ can be regarded as a subspace of continuous functions on $A^*$ with the weak* topology, and any such measure $\mu_F$ can be regarded as a regular Borel measure defined on a closed subset of $A^*$."

I'm having difficult to understand it.

In the first statement, I came to the conclusion that he reffers to the image of $A$ by the James map $J: A \to A^{**}$, and since the weak* topology preserves the continuity of the functionals in $J(A)$, every element of $J(A)$ is a continuous function on the topological space $(A^*,\omega^*)$. The map $J$ is linear (so $J(A)$ is a subspace of $(A^*,\omega^*)$).

I guess there is a second possibility to interpret the first statement, that is to consider the map $\delta: X \to A^*$ such that $\delta(x)$ is the evaluation map on $x$ restricted to $A$. But then I don't know where to go from here.

In the second statement, I have no idea what meant. Please help me.