  The property of almost every point being a Lebesgue point has proven to be
crucial for the consistency of several classification algorithms based on
nearest neighbors. We characterize Lebesgue points in terms of a 1-Nearest
Neighbor regression algorithm for pointwise estimation, fleshing out the role
played by tie-breaking rules in the corresponding convergence problem. We then
give an application of our results, proving the convergence of the risk of a
large class of 1-Nearest Neighbor classification algorithms in general metric
spaces where almost every point is a Lebesgue point.
