A nearest neighbor characterization of Lebesgue points in metric measure spaces

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.