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：本文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年）及Preiss（1983年）的研究成果，该规则在Nagata意义下所有$\sigma$有限维度量空间$X$中已知具有一致收敛性。本文证明，在无平局情形下，此类空间中的$k$-NN规则具有强一致收敛性。结合Devroye、Györfi、Krzyżak与Lugosi（1994年）在欧氏场景中采用的破平局策略，我们成功证明了该规则在非阿基米德度量空间（即Nagata维数为零的空间）中的强一致收敛性。通过将Cérou与Guyader定理与Assouad及Quentin de Gromard（2006年）的结论相结合，可推知$k$-NN规则在de Groot意义下具有有限维数的度量空间中具有一致收敛性。特别地，该规则在海森堡群中具有一致收敛性——尽管该空间在Nagata意义下并非$\sigma$有限维（该反例由Korányi与Reimann（1995年）及Sawyer与Wheeden（1992年）独立构造）。