A simple proof of Singer's representation theorem

Let $\Omega$ be a compact Hausdorff space and $X$ a Banach space. Singer's theorem states that under the dual pairing $(f,m)\mapsto \int \langle f,dm\rangle$, the dual space of $C(\Omega ;X)$ is isometric to $rcabv (\Omega ;X')$. Using the Hahn-Banach theorem and the (scalar) Riesz representation theorem, a proof of Singer's theorem is given which appears to be simpler than the proofs supplied earlier by Singer (1957, 1959) and Dinculeanu (1959, 1967).