  This paper characterizes the maximum mean discrepancies (MMD) that metrize
the weak convergence of probability measures for a wide class of kernels. More
precisely, we prove that, on a locally compact, non-compact, Hausdorff space,
the MMD of a bounded continuous Borel measurable kernel k, whose reproducing
kernel Hilbert space (RKHS) functions vanish at infinity, metrizes the weak
convergence of probability measures if and only if k is continuous and
integrally strictly positive definite (i.s.p.d.) over all signed, finite,
regular Borel measures. We also correct a prior result of Simon-Gabriel &
Sch\"olkopf (JMLR, 2018, Thm.12) by showing that there exist both bounded
continuous i.s.p.d. kernels that do not metrize weak convergence and bounded
continuous non-i.s.p.d. kernels that do metrize it.
