We prove the following analogue of the Riesz-Representation theorem for the space of quaternion-valued continuous functions on a compact Hausdorff space: Let $X$ be a compact Hausdorff space and $\psi\colon C(X,\mathbb H \to\mathbb H$ be a bounded linear functional on a left quaternion normed linear space $C(X,\mathbb H)$, then there exists a unique quaternion valued regular Borel measure $\lambda$ on the $\sigma$-algebra of all Borel subsets of $X$ such that \[ si(f)=ıt_xf\,dx,ext{ for all }fı C(X,\mathbb H) \] and $\|\psi\|=|\lambda|(X)$, $|\lambda|$ is the total variation of $\lambda$. Some basic results (needed in the proof of the main theorem) from the theory of quaternion measures are also proved. These include an analogue of Lusin's theorem and an analogue of the Radon-Nikodym theorem.