Luzin N property

Remarks

Example: a Lipschitz function $f$ with constant $L$ has the property. Let $\lambda(E) = 0$ and $\varepsilon > 0$, and cover $E$ by intervals $I_k$ of total length less than $\varepsilon$. Each set $f(E \cap I_k)$ lies in an interval of length at most $L \cdot \abs{I_k}$. So $f(E)$ is covered by intervals of total length less than $L \varepsilon$.

Non-example: the Cantor function maps the Cantor set $C$, which has measure $0$, onto $[0, 1]$, which has measure $1$.

The image $f(E)$ need not be measurable for a general $f$, so the condition is read with Lebesgue outer measure. A continuous function on $[a, b]$ has the N property if and only if it maps Lebesgue measurable sets to Lebesgue measurable sets.