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'' "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" />''
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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.
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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.
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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" />.
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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.
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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).
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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.
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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" />.
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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.
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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" />.
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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.
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====References====
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<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>
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====Comments====
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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
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<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>
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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]]].
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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
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="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" />,
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<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>
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(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
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<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>
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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" />.
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====References====
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<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>
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Revision as of 07:39, 7 December 2012

"null-propertynull-property" , of a function , continuous on an interval

For any set of measure , the image of this set, , also has measure zero. It was introduced by N.N. Luzin in 1915 (see [1]). The following assertions hold.

1) A function on such that almost-everywhere on does not have the Luzin -property.

2) If does not have the Luzin -property, then on there is a perfect set of measure zero such that .

3) An absolutely continuous function has the Luzin -property.

4) If has the Luzin -property and has bounded variation on (as well as being continuous on ), then is absolutely continuous on (the Banach–Zaretskii theorem).

5) If does not decrease on and is finite on , then has the Luzin -property.

6) In order that be measurable for every measurable set it is necessary and sufficient that have the Luzin -property on .

7) A function that has the Luzin -property has a derivative on the set for which any non-empty portion of it has positive measure.

8) For any perfect nowhere-dense set there is a function having the Luzin -property on and such that does not exist at any point of .

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


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Nikita2/sandbox. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Nikita2/sandbox&oldid=29106