  We show that a radial continuous valuation defined on the $n$-dimensional
star bodies extends uniquely to a continuous valuation on the $n$-dimensional
bounded star sets. Moreover, we provide an integral representation of every
such valuation, in terms of the radial function, which is valid on the dense
subset of the simple Borel star sets. We also show that every radial continuous
valuation defined on the $n$-dimensional star bodies can be decomposed as a sum
$V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous
valuations.
