# A Nearest Neighbor Characterization of Lebesgue Points in Metric Measure Spaces

arXiv: https://arxiv.org/abs/2007.03937 · HTML: https://mlsanity.com/paper/2007.03937/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2020.

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## Abstract (arXiv)

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.