# Luzin-N-property

"null-propertynull-property" , of a function $f$, continuous on an interval $[ a , b ]$

For any set $E \subset [ a , b ]$ of measure $\mathop{\rm mes} E = 0$, the image of this set, $f ( E)$, also has measure zero. It was introduced by N.N. Luzin in 1915 (see ). The following assertions hold.

1) A function $f \not\equiv \textrm{ const }$ on $[ a , b ]$ such that $f ^ { \prime } ( x) = 0$ almost-everywhere on $[ a , b ]$ does not have the Luzin $N$- property (for example Cantor ternary function).

2) If $f$ does not have the Luzin $N$- property, then on $[ a , b ]$ there is a perfect set $P$ of measure zero such that $\mathop{\rm mes} f ( P) > 0$.

3) An absolutely continuous function has the Luzin $N$- property.

4) If $f$ has the Luzin $N$- property and has bounded variation on $[ a , b ]$( as well as being continuous on $[ a , b ]$), then $f$ is absolutely continuous on $[ a , b ]$( the Banach–Zaretskii theorem).

5) If $f$ does not decrease on $[ a , b ]$ and $f ^ { \prime }$ is finite on $[ a , b ]$, then $f$ has the Luzin $N$- property.

6) In order that $f ( E)$ be measurable for every measurable set $E \subset [ a , b ]$ it is necessary and sufficient that $f$ have the Luzin $N$- property on $[ a , b ]$.

7) A function $f$ that has the Luzin $N$- property has a derivative $f ^ { \prime }$ on the set for which any non-empty portion of it has positive measure.

8) For any perfect nowhere-dense set $P \subset [ a , b ]$ there is a function $f$ having the Luzin $N$- property on $[ a , b ]$ and such that $f ^ { \prime }$ does not exist at any point of $P$.

The concept of Luzin's $N$- property can be generalized to functions of several variables and functions of a more general nature, defined on measure spaces.

How to Cite This Entry:
Luzin-N-property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin-N-property&oldid=47720
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article