  We consider smoothed versions of geometric range spaces, so an element of the
ground set (e.g. a point) can be contained in a range with a non-binary value
in $[0,1]$. Similar notions have been considered for kernels; we extend them to
more general types of ranges. We then consider approximations of these range
spaces through $\varepsilon $-nets and $\varepsilon $-samples (aka
$\varepsilon$-approximations). We characterize when size bounds for
$\varepsilon $-samples on kernels can be extended to these more general
smoothed range spaces. We also describe new generalizations for $\varepsilon
$-nets to these range spaces and show when results from binary range spaces can
carry over to these smoothed ones.
