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)

Comments

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