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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a function.\nWrite $\\lambda$ for [[Lebesgue measure|2Z2QVWZ262]] on $\\R$.\nThe function $f$ has the *Luzin N property* if for every set $E \\subseteq [a, b]$ with $\\lambda(E) = 0$, the image $f(E)$ satisfies $\\lambda(f(E)) = 0$.\n\n## Remarks\n\nExample: a [[Lipschitz function|0QNVQ4VD9T]] $f$ with constant $L$ has the property.\nLet $\\lambda(E) = 0$ and $\\varepsilon \u003e 0$, and cover $E$ by intervals $I_k$ of total length less than $\\varepsilon$.\nEach set $f(E \\cap I_k)$ lies in an interval of length at most $L \\cdot \\abs{I_k}$.\nSo $f(E)$ is covered by intervals of total length less than $L \\varepsilon$.\n\nNon-example: the [[Cantor function|NPBYYK63FE]] maps the Cantor set $C$, which has measure $0$, onto $[0, 1]$, which has measure $1$.\n\nThe image $f(E)$ need not be measurable for a general $f$, so the condition is read with Lebesgue outer measure.\nA continuous function on $[a, b]$ has the N property if and only if it maps Lebesgue measurable sets to Lebesgue measurable sets.\n",
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