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  "body": "\n## Summary\n\nA function has the Luzin N property when it maps every set of Lebesgue measure zero to a set of Lebesgue measure zero.\nThe property singles out the [[absolutely continuous|ZT6F4C2YHK]] functions among the continuous functions of [[bounded variation|A8251BHCMP]], by the [[Banach–Zarecki theorem|ZMEYSAHNX3]].\nIt also appears in the change of variables formula for Lebesgue integrals and in the study of Sobolev maps.\n\n## Explanations\n\n### L1 human\n\nA [[null set|VQQM4Y550T]] is a set so thin that it has length zero, such as a countable set or the [[Cantor set|0BXNDZFQME]].\nA function with the N property never blows up such a thin set into a set of positive length.\nEvery Lipschitz function has the property, because it stretches lengths by at most a fixed factor.\nThe [[Cantor function|NPBYYK63FE]] does not: it maps the Cantor set, of length $0$, onto the whole interval $[0, 1]$.\n",
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  "msc": [
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  "name": "Luzin N property",
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