Subsampling in Smoothed Range Spaces

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 ε-nets and εsamples (aka ε-approximations). We characterize when size bounds for ε-samples on kernels can be extended to these more general smoothed range spaces. We also describe new generalizations for ε-nets to these range spaces and show when results from binary range spaces can carry over to these smoothed ones.