---
tag: WMHMGJD261
kind: concept
name: Luzin N property
msc: [26A46, 26A30]
sources:
  - url: https://en.wikipedia.org/wiki/Luzin_N_property
same_as:
  wikidata: Q1160012
  wikipedia: Luzin N property
---

## Summary

A function has the Luzin N property when it maps every set of Lebesgue measure zero to a set of Lebesgue measure zero.
The property singles out the [[absolutely continuous|ZT6F4C2YHK]] functions among the continuous functions of [[bounded variation|A8251BHCMP]], by the [[Banach–Zarecki theorem|ZMEYSAHNX3]].
It also appears in the change of variables formula for Lebesgue integrals and in the study of Sobolev maps.

## Explanations

### L1 human

A [[null set|VQQM4Y550T]] is a set so thin that it has length zero, such as a countable set or the [[Cantor set|0BXNDZFQME]].
A function with the N property never blows up such a thin set into a set of positive length.
Every Lipschitz function has the property, because it stretches lengths by at most a fixed factor.
The [[Cantor function|NPBYYK63FE]] does not: it maps the Cantor set, of length $0$, onto the whole interval $[0, 1]$.