  We show that the $k$-nearest neighbour learning rule is universally
consistent in a metric space $X$ if and only if it is universally consistent in
every separable subspace of $X$ and the density of $X$ is less than every
real-measurable cardinal. In particular, the $k$-NN classifier is universally
consistent in every metric space whose separable subspaces are sigma-finite
dimensional in the sense of Nagata and Preiss if and only if there are no
real-valued measurable cardinals. The latter assumption is relatively
consistent with ZFC, however the consistency of the existence of such cardinals
cannot be proved within ZFC. Our results were inspired by an example sketched
by C\'erou and Guyader in 2006 at an intuitive level of rigour.
