Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. III

We establish the last missing link allowing to describe those complete separable metric spaces $X$ in which the $k$ nearest neighbour classifier is universally consistent, both in combinatorial terms of dimension theory and via a fundamental property of real analysis. The following are equivalent: (1) The $k$-nearest neighbour classifier is universally consistent in $X$, (2) The strong Lebesgue--Besicovitch differentiation property holds in $X$ for every locally finite Borel measure, (3) $X$ is sigma-finite dimensional in the sense of Jun-Iti Nagata. The equivalence (2)$\iff$(3) was announced by Preiss (1983), while a detailed proof of the implication (3)$\Rightarrow$(2) has only appeared in Assouad and Quentin de Gromard (2006). The implication (2)$\Rightarrow$(1) was established by Cérou and Guyader (2006). We prove the implication (1)$\Rightarrow$(3). We further show that the weak (instead of strong) Lebesgue--Besicovitch property is insufficient for the consistency of the $k$-NN rule, as witnessed, for example, by the Heisenberg group (here we correct a wrong claim made in the previous article (Kumari and Pestov 2024)). A bit counter-intuitively, there is a metric on the real line uniformly equivalent to the usual distance but under which the $k$-NN classifier fails. Finally, another equivalent condition that can be added to the above is the Cover--Hart property: (4) the error of the $1$-nearest neighbour classifier is asymptotically at most twice as bad as the Bayes error.