Theorem 12 of Simon-Gabriel & Sch\"olkopf (JMLR, 2018) seemed to close a 40-year-old quest to characterize maximum mean discrepancies (MMD) that metrize the weak convergence of probability measures. We prove, however, that the theorem is incorrect and provide a correction. We show that, on a locally compact, non-compact, Hausdorff space, the MMD of a bounded continuous Borel measurable kernel k, whose RKHS-functions vanish at infinity, metrizes the weak convergence of probability measures if and only if k is continuous and integrally strictly positive definite (ISPD) over all signed, finite, regular Borel measures. We also show that, contrary to the claim of the aforementioned Theorem 12, there exist both bounded continuous ISPD kernels that do not metrize weak convergence and bounded continuous non-ISPD kernels that do metrize it.

翻译：Simon-Gabriel & Sch\'olkopf(JMLR,2018年)的12号理论似乎结束了40年来对最大平均值差异(MMD)的描述,这种差异使概率措施的趋同弱化。然而,我们证明,该理论是不正确的,并提供了纠正。我们证明,在当地一个非集约、非集约的Hausdorf空间,一个受约束的连续连续波雷尔可测量内核的MDMD(MMD)中,该内核的RKHS功能在无限性时消失,使概率措施的趋同弱(MMD)在K是持续和完全确定(ISPD)对所有已签署的、限定的、常规的博雷尔措施(ISPD)中微弱的趋同。我们还表明,与上述12号理论的主张相反,存在两个不协调弱趋同的连续的ISD内核内核的相互连接的内核,没有使弱趋同和连续的非ISPD内核内核内核的内核进行融化。