On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found:
Let $(X, d)$ be a compact metric. Recall that the dual space $C(X)^*$ of continuous real functionals on $C(X)$ of continuous functions $X\to \mathbf R$ can be naturally identified with the space of finite signed measures on $X$ equipped with the weak* topology.
I want to confirm if the correspondence is given by the following map:
Let $\mathcal F$ be the space of all the finite signed measures on $X$. Define $\phi:\mathcal F\to C(X)^*$ as $$\phi(\mu) = (f\mapsto \int f\ d\mu)$$