Namespaces
Variants
Actions

Difference between revisions of "Luzin-N-property"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (→‎Comments: typo)
m (tex encoded by computer)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
'' "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" />''
+
<!--
 +
l0610502.png
 +
$#A+1 = 88 n = 0
 +
$#C+1 = 88 : ~/encyclopedia/old_files/data/L061/L.0601050 Luzin \BMI N\EMI\AAhproperty,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
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.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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.
+
'' "null-propertynull-property" , of a function  $  f $,
 +
continuous on an interval  $  [ a , b ] $''
  
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" />.
+
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 [[#References|[1]]]). The following assertions hold.
  
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.
+
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]]).
  
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).
+
2) If $  f $
 +
does not have the Luzin $  N $-
 +
property, then on $  [ a , b ] $
 +
there is a [[Perfect set|perfect set]]  $  P $
 +
of measure zero such that  $  \mathop{\rm mes}  f ( P) > 0 $.
  
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.
+
3) An absolutely continuous function has the Luzin $  N $-
 +
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" />.
+
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).
  
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.
+
5) If  $  f $
 +
does not decrease on  $  [ a , b ] $
 +
and  $  f ^ { \prime } $
 +
is finite on  $  [ a , b ] $,
 +
then  $  f $
 +
has the Luzin $  N $-
 +
property.
  
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" />.
+
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 ] $.
  
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.
+
7) A function  $  f $
 +
that has the Luzin  $  N $-
 +
property has a derivative  $  f ^ { \prime } $
 +
on the set for which any non-empty [[Portion|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====
 
====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>
 
<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====
 
====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 all Lebesgue-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105050.png" />,
+
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 $,
  
<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>
+
$$
 +
\mathop{\rm mes} ( E) < \delta  \Rightarrow  \mathop{\rm mes} ( f ( E) ) < \epsilon .
 +
$$
  
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]]].
+
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 [[#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" />,
+
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 $  \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 ) $,
  
<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>
+
$$
 +
P ( E) = 0  \Rightarrow  \lambda ( F ( E) ) = 0
 +
$$
  
(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
+
(respectively, for all $  \epsilon > 0 $
 +
there is a $  \delta > 0 $
 +
such that for all $  E \in {\mathcal B} ( \Omega ) $
 +
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>
+
$$
 +
P ( E) < \delta  \Rightarrow  \lambda ( F ( E) ) < \epsilon \textrm{ )  } .
 +
$$
  
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" />.
+
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====
 
====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>
 
<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>

Latest revision as of 04:11, 6 June 2020


"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
How to Cite This Entry:
Luzin-N-property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin-N-property&oldid=36813
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article