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 [1]). 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.

References

 [1] N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1915) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)

There is another property intimately related to the Luzin $N$- property. A function $f$ continuous on an interval $[ a , b ]$ has the Banach $S$- property if for all $E \subset [ a, b ]$ there exists a $\epsilon > 0$ such that for all Lebesgue-measurable sets $\delta > 0$,

$$\mathop{\rm mes} ( E) < \delta \Rightarrow \mathop{\rm mes} ( f ( E) ) < \epsilon .$$

This is clearly stronger than the $N$- property. S. Banach proved that a function $f$ has the $S$- property (respectively, the $N$- property) if and only if (respectively, only if — see below for the missing "if" ) the inverse image $f ^ { - 1 } ( \{ x \} )$ is finite (respectively, is at most countable) for almost-all $x$ in $f ( [ a , b ] )$. For classical results on the $N$- and $S$- properties, see [a3].

Recently a powerful extension of these results has been given by G. Mokobodzki (cf. [a1], [a2]), allowing one to prove deep results in potential theory. Let $\Omega$ and $T$ be two compact metrizable spaces, $\Omega$ being equipped with a probability measure $P$. Let $F$ be a Borel subset of $\Omega \times T$ and, for any Borel subset $E$ of $\Omega$, define the subset $F ( E)$ of $T$ by $F ( E) = \{ {t \in T } : {\textrm{ there is an } \omega \in \Omega \textrm{ such that } ( \omega , t ) \in F } \}$( if $F$ is the graph of a mapping $f: \Omega \rightarrow T$, then $F ( E) = f ( E)$). The set $F$ is said to have the property (N) (respectively, the property (S)) if there exists a measure $\lambda$ on $T$( here depending on $F$) such that for all $E \in {\mathcal B} ( \Omega )$,

$$P ( E) = 0 \Rightarrow \lambda ( F ( E) ) = 0$$

(respectively, for all $\epsilon > 0$ there is a $\delta > 0$ such that for all $E \in {\mathcal B} ( \Omega )$ one has

$$P ( E) < \delta \Rightarrow \lambda ( F ( E) ) < \epsilon \textrm{ ) } .$$

Now $F$ has the property (N) (respectively, the property (S)) if and only if the section $F ( \omega )$ of $F$ is at most countable (respectively, is finite) for almost-all $\omega \in \Omega$.

References

 [a1] C. Dellacherie, D. Feyel, G. Mokobodzki, "Intégrales de capacités fortement sous-additives" , Sem. Probab. Strasbourg XVI , Lect. notes in math. , 920 , Springer (1982) pp. 8–28 MR0658670 Zbl 0496.60076 [a2] A. Louveau, "Minceur et continuité séquentielle des sous-mesures analytiques fortement sous-additives" , Sem. Initiation à l'Analyse , 66 , Univ. P. et M. Curie (1983–1984) Zbl 0587.28003 [a3] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05 [a4] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
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