We continue to investigate the $k$ nearest neighbour learning rule in separable metric spaces. Thanks to the results of C\'erou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Gy\"{o}rfi, Krzy\.{z}ak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of C\'erou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the $k$-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the $k$-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Kor\'anyi and Reimann (1995) and Sawyer and Wheeden (1992).

翻译：我们继续研究可分度量空间中的$k$近邻学习规则。基于Cérou与Guyader（2006）以及Preiss（1983）的结论，该规则在每一位长田意义下$\sigma$有限维的度量空间$X$中均被证明具有普适一致性。本文进一步证明，在无平局情形下，此类空间中该规则具有强普适一致性。针对Devroye、Győrfi、Krzyżak与Lugosi（1994）在欧氏设定下采用的破平局策略，我们在非阿基米德度量空间（即长田维数为零的空间）中成功证明了强普适一致性。结合Cérou与Guyader的定理与Assouad与Quentin de Gromard（2006）的结果，可推知$k$-近邻规则在de Groot意义下具有有限维的度量空间中具有普适一致性。特别地，该规则在海森堡群中具有普适一致性——该群并非长田意义下的$\sigma$有限维空间，这一点可分别由Korányi与Reimann（1995）以及Sawyer与Wheeden（1992）独立构造的反例证实。