 Riesz–Markov–Kakutani representation theorem

Every positive linear functional $\phi$ on $C(K)$ for a compact Hausdorff space $K$ has a unique representationby a finite regular positive Borel measure $\mu$. More generally, $C(K{)}^{\ast }$ is isometrically the space of finite regular complex Borel measures with the total-variation norm.

The Riesz-Markov-Kakutani representation theorem is a fundamental result in measure theory and functional analysis, particularly in the context of representing positive linear functionals on spaces of continuous functions. It provides a powerful method to characterize and represent certain types of measures through continuous functions.