  The $k$ nearest neighbour learning rule (under the uniform distance tie
breaking) is universally consistent in every metric space $X$ that is
sigma-finite dimensional in the sense of Nagata. This was pointed out by
C\'erou and Guyader (2006) as a consequence of the main result by those
authors, combined with a theorem in real analysis sketched by D. Preiss (1971)
(and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show
that it is possible to give a direct proof along the same lines as the original
theorem of Charles J. Stone (1977) about the universal consistency of the
$k$-NN classifier in the finite dimensional Euclidean space. The generalization
is non-trivial because of the distance ties being more prevalent in the
non-euclidean setting, and on the way we investigate the relevant geometric
properties of the metrics and the limitations of the Stone argument, by
constructing various examples.
