We continue to investigate the $k$ nearest neighbour ($k$-NN) learning rule in complete 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 such metric space 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：我们继续研究完备可分度量空间中的$k$近邻学习规则。基于Cérou与Guyader（2006）及Preiss（1983）的研究成果，该规则已在所有Nagata意义下σ有限维的度量空间中被证明具有普适一致性。本文证明，在无平局情形下，该规则在此类空间中具有强普适一致性。针对Devroye、Győrfi、Krżyżak与Lugosi（1994）在欧氏空间中采用的破平局策略，我们成功证明了该策略在非阿基米德度量空间（即Nagata维度为零的空间）中具有强普适一致性。结合Cérou与Guyader定理以及Assouad与Quentin de Gromard（2006）的结论，可推导出在de Groot意义下有限维的度量空间中，$k$-NN规则具有普适一致性。特别地，$k$-NN规则在海森堡群中具有普适一致性——该群并非Nagata意义下的σ有限维空间，此结论源于Korányi与Reimann（1995）以及Sawyer与Wheeden（1992）独立构造的反例。