  Strictly proper kernel scores are well-known tool in probabilistic
forecasting, while characteristic kernels have been extensively investigated in
the machine learning literature. We first show that both notions coincide, so
that insights from one part of the literature can be used in the other. We then
show that the metric induced by a characteristic kernel cannot reliably
distinguish between distributions that are far apart in the total variation
norm as soon as the underlying space of measures is infinite dimensional. In
addition, we provide a characterization of characteristic kernels in terms of
eigenvalues and -functions and apply this characterization to the case of
continuous kernels on (locally) compact spaces. In the compact case we further
show that characteristic kernels exist if and only if the space is metrizable.
As special cases of our general theory we investigate translation-invariant
kernels on compact Abelian groups and isotropic kernels on spheres. The latter
are of particular interest for forecast evaluation of probabilistic predictions
on spherical domains as frequently encountered in meteorology and climatology.
