Let $K$ be a [compact](/page/Compact%20Space) Hausdorff space. For every $f \in C(K)^*$, there exists a unique finite signed [Borel](/page/Borel%20Sigma%20Algebra) regular measure $\mu_f$ on $K$ such that \begin{align*} f(\phi) = \int_K \phi \, d\mu_f \quad \text{for all } \phi \in C(K). \end{align*} The map $f \mapsto \mu_f$ is an isometric isomorphism $C(K)^* \cong \mathcal{M}(K)$, where $\mathcal{M}(K)$ is the space of finite signed Borel regular measures with total variation norm $\|\mu\|_{\mathcal{M}} = |\mu|(K)$.