  We characterize the positive radial continuous and rotation invariant
valuations $V$ defined on the star bodies of $\mathbb R^n$ as the applications
on star bodies which admit an integral representation with respect to the
Lebesgue measure. That is, $$V(K)=\int_{S^{n-1}}\theta(\rho_K)dm,$$ where
$\theta$ is a positive continuous function, $\rho_K$ is the radial function
associated to $K$ and $m$ is the Lebesgue measure on $S^{n-1}$. As a corollary,
we obtain that every such valuation can be uniformly approximated on bounded
sets by a linear combination of dual quermassintegrals.
