# On Lebesgue null sets

## TL;DR

This paper characterizes Lebesgue null sets in Euclidean space by examining their intersections with Lipschitz-varying affine subspaces and establishing an equivalence involving Hausdorff measure.

## Contribution

It provides a new criterion for Lebesgue null sets based on their intersection properties with affine subspaces that vary Lipschitz continuously.

## Key findings

- A Borel set is Lebesgue null iff its intersection with almost every affine subspace has zero Hausdorff measure.
- The result links measure-theoretic nullity to geometric intersection properties.
- The approach uses Lipschitz variation of affine subspaces to characterize null sets.

## Abstract

Letting A be a Borel subset of n dimensional Euclidean space, and W(x) be an m dimensional affine subspace containing x and varying in a Lipschitz way according to x, we establish that A is Lebesgue null if and only if $A \cap W(x)$ has m dimensional Hausdorff measure vanishing for almost every x.