View source for Luzin-N-property

'' "null-propertynull-property" , of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610502.png" />, continuous on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610503.png" />''

For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610504.png" /> of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610505.png" />, the image of this set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610506.png" />, also has measure zero. It was introduced by N.N. Luzin in 1915 (see [[#References|[1]]]). The following assertions hold.

1) A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610507.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610508.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l0610509.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105010.png" /> does not have the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105011.png" />-property.

2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105012.png" /> does not have the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105013.png" />-property, then on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105014.png" /> there is a [[Perfect set|perfect set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105015.png" /> of measure zero such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105016.png" />.

3) An absolutely continuous function has the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105017.png" />-property.

4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105018.png" /> has the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105019.png" />-property and has bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105020.png" /> (as well as being continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105021.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105022.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105023.png" /> (the Banach–Zaretskii theorem).

5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105024.png" /> does not decrease on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105026.png" /> is finite on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105028.png" /> has the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105029.png" />-property.

6) In order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105030.png" /> be measurable for every measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105031.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105032.png" /> have the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105033.png" />-property on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105034.png" />.

7) A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105035.png" /> that has the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105036.png" />-property has a derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105037.png" /> on the set for which any non-empty [[Portion|portion]] of it has positive measure.

8) For any perfect nowhere-dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105038.png" /> there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105039.png" /> having the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105040.png" />-property on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105041.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105042.png" /> does not exist at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105043.png" />.

The concept of Luzin's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105044.png" />-property can be generalized to functions of several variables and functions of a more general nature, defined on measure spaces.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,   "The integral and trigonometric series" , Moscow-Leningrad  (1915)  (In Russian)  (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)  {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====
There is another property intimately related to the Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105045.png" />-property. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105046.png" /> continuous on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105047.png" /> has the Banach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105049.png" />-property if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105051.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105052.png" /> such that for allLebesgue-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105050.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105053.png" /></td> </tr></table>

This is clearly stronger than the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105054.png" />-property. S. Banach proved that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105055.png" /> has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105056.png" />-property (respectively, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105057.png" />-property) if and only if (respectively, only if — see below for the missing  "if" ) the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105058.png" /> is finite (respectively, is at most countable) for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105060.png" />. For classical results on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105061.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105062.png" />-properties, see [[#References|[a3]]].

Recently a powerful extension of these results has been given by G. Mokobodzki (cf. [[#References|[a1]]], [[#References|[a2]]]), allowing one to prove deep results in potential theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105064.png" /> be two compact metrizable spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105065.png" /> being equipped with a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105066.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105067.png" /> be a Borel subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105068.png" /> and, for any Borel subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105070.png" />, define the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105072.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105073.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105074.png" /> is the graph of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105076.png" />). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105077.png" /> is said to have the property (N) (respectively, the property (S)) if there exists a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105079.png" /> (here depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105080.png" />) such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105081.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105082.png" /></td> </tr></table>

(respectively, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105083.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105084.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105085.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105086.png" /></td> </tr></table>

Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105087.png" /> has the property (N) (respectively, the property (S)) if and only if the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105089.png" /> is at most countable (respectively, is finite) for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105090.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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  {{MR|0658670}} {{ZBL|0496.60076}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)  {{MR|}} {{ZBL|0587.28003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)  {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}}  {{ZBL|63.0183.05}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR></table>