Difference between revisions of "User:Nikita2/sandbox"
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+ | A mapping $\varphi:D\to D'$ possesses Luzin's $\mathcal N$-property if the image of every set of measure zero is a set of measure zero. A mapping $\varphi$ possesses Luzin's $\mathcal N{}^{-1}$-property if the preimage of every set of measure zero is a set of measure zero. | ||
+ | |||
+ | ''Briefly'' | ||
+ | \begin{equation*} | ||
+ | \mathcal N\text{-property:}\quad \Sigma\subset D, |\Sigma| = 0 \Rightarrow |\varphi(\Sigma)|=0, | ||
+ | \end{equation*} | ||
+ | \begin{equation*} | ||
+ | \mathcal N{}^{-1}\text{-property:} \quad M \subset D, |M| = 0 \Rightarrow |\varphi^{-1}(M)|=0. | ||
+ | \end{equation*} | ||
+ | |||
+ | |||
+ | ====$\mathcal N$-property of a function $f$ on an interval $[a,b]$==== | ||
+ | |||
+ | Let $f:[a,b]\to \mathbb R$ be a measurable function. In this case the definition is following: | ||
+ | ''For any set $E\subset[a,b]$ of measure zero ($|E|=0$), 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. | ||
+ | # A function $f\not\equiv \operatorname{const}$ on $[a,b]$ such that $f'(x)=0 $ almost-everywhere on $[a,b]$ (see for example [[Cantor_ternary_function|Cantor ternary function]]) does not have the Luzin $\mathcal N$-property. | ||
+ | # If $f$ does not have the Luzin $\mathcal N$-property, then on $[a,b]$ there is a [[Perfect set|perfect set]] $P$ of measure zero such that $|f(P)|>0$. | ||
+ | # An [[Absolutely_continuous_function#Absolute_continuity_of_a_function|absolutely continuous function]] has the Luzin $\mathcal N$-property. | ||
+ | # If $f$ has the Luzin $\mathcal N$-property and has [[Function_of_bounded_variation#Classical_definition|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). | ||
+ | # If $f$ does not decrease on $[a,b]$ and $f'$ is finite on $[a,b]$, then $f$ has the Luzin $\mathcal N$-property. | ||
+ | # 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 $\mathcal N$-property on $[a,b]$. | ||
+ | # A function $f$ that has the Luzin $\mathcal N$-property has a derivative $f'$ on the set for which any non-empty [[Portion|portion]] of it has positive measure. | ||
+ | # For any perfect nowhere-dense set $P\subset[a,b]$ there is a function $f$ having the Luzin $\mathcal N$-property on $[a,b]$ and such that $f'$ does not exist at any point of $P$. | ||
+ | |||
+ | 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 Lebesgue-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105050.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105051.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061050/l06105052.png" /> such that | ||
+ | |||
+ | <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> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | ===AsymptotePlay=== | ||
<asy> | <asy> | ||
import graph; | import graph; | ||
− | |||
− | |||
− | + | size(400,200,IgnoreAspect); | |
− | + | ||
− | + | int n=2; | |
− | + | ||
− | + | real[] a=new real[n]; | |
+ | real[] x=new real[n]; | ||
− | + | //real[] x=sequence(n); | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
+ | string[] month={"Jan","February","March","April","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"}; | ||
− | + | a[0]=349; //February | |
− | + | a[1]=353; //March | |
− | + | a[2]=363; //April | |
− | + | a[3]=353; //May | |
+ | a[4]=353; //June | ||
− | + | void mbox(real m, real c) | |
− | + | { | |
− | + | path g=box((m-0.5,0),(m+0.5,c)); | |
− | + | filldraw(g,lightgrey); | |
− | + | } | |
− | + | for(int i=0; i<n;++i){x[i]=i+0.5;} | |
− | + | for(int i=0; i < n; ++i){ mbox(i,a[i]);} | |
− | + | //draw((-0.5,300)--(n-0.5,300)); | |
+ | draw(box((-0.5,300),(n-0.5,a[n]+2))); | ||
+ | limits((-0.5,300),(n-0.5,a[n]+2),Crop); | ||
− | + | xaxis(xmin=0,Bottom,LeftTicks(Step=1,new string(real y) {return month[round(y+1 % 12)];})); | |
+ | yaxis("pages with TeX",Left,LeftTicks(DefaultFormat, new real[] {300,a[0],a[1]})); | ||
</asy> | </asy> | ||
+ | |||
+ | ===FigurePlay=== | ||
+ | <span id="Fig1", bg="black"> | ||
+ | [[File:Cochleoid-1.png| right| frame| Figure 1. The cochleoid ([[Media:Cochleoid-1.pdf|pdf]]) ]] | ||
+ | </span> |
Latest revision as of 07:08, 3 March 2013
A mapping $\varphi:D\to D'$ possesses Luzin's $\mathcal N$-property if the image of every set of measure zero is a set of measure zero. A mapping $\varphi$ possesses Luzin's $\mathcal N{}^{-1}$-property if the preimage of every set of measure zero is a set of measure zero.
Briefly \begin{equation*} \mathcal N\text{-property:}\quad \Sigma\subset D, |\Sigma| = 0 \Rightarrow |\varphi(\Sigma)|=0, \end{equation*} \begin{equation*} \mathcal N{}^{-1}\text{-property:} \quad M \subset D, |M| = 0 \Rightarrow |\varphi^{-1}(M)|=0. \end{equation*}
$\mathcal N$-property of a function $f$ on an interval $[a,b]$
Let $f:[a,b]\to \mathbb R$ be a measurable function. In this case the definition is following: For any set $E\subset[a,b]$ of measure zero ($|E|=0$), 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.
- A function $f\not\equiv \operatorname{const}$ on $[a,b]$ such that $f'(x)=0 $ almost-everywhere on $[a,b]$ (see for example Cantor ternary function) does not have the Luzin $\mathcal N$-property.
- If $f$ does not have the Luzin $\mathcal N$-property, then on $[a,b]$ there is a perfect set $P$ of measure zero such that $|f(P)|>0$.
- An absolutely continuous function has the Luzin $\mathcal N$-property.
- If $f$ has the Luzin $\mathcal N$-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).
- If $f$ does not decrease on $[a,b]$ and $f'$ is finite on $[a,b]$, then $f$ has the Luzin $\mathcal N$-property.
- 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 $\mathcal N$-property on $[a,b]$.
- A function $f$ that has the Luzin $\mathcal N$-property has a derivative $f'$ on the set for which any non-empty portion of it has positive measure.
- For any perfect nowhere-dense set $P\subset[a,b]$ there is a function $f$ having the Luzin $\mathcal N$-property on $[a,b]$ and such that $f'$ does not exist at any point of $P$.
The concept of Luzin's -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 -property. A function
continuous on an interval
has the Banach
-property if for all Lebesgue-measurable sets
and all
is a
such that
![]() |
This is clearly stronger than the -property. S. Banach proved that a function
has the
-property (respectively, the
-property) if and only if (respectively, only if — see below for the missing "if" ) the inverse image
is finite (respectively, is at most countable) for almost-all
in
. For classical results on the
- and
-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 and
be two compact metrizable spaces,
being equipped with a probability measure
. Let
be a Borel subset of
and, for any Borel subset
of
, define the subset
of
by
(if
is the graph of a mapping
, then
). The set
is said to have the property (N) (respectively, the property (S)) if there exists a measure
on
(here depending on
) such that for all
,
![]() |
(respectively, for all there is a
such that for all
one has
![]() |
Now has the property (N) (respectively, the property (S)) if and only if the section
of
is at most countable (respectively, is finite) for almost-all
.
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 |
AsymptotePlay
FigurePlay
![](/images/5/56/Cochleoid-1.png)
Nikita2/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikita2/sandbox&oldid=29096