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''for measurability of a function of a real variable''
 
''for measurability of a function of a real variable''
  
For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610101.png" />, defined on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610102.png" /> and almost-everywhere finite, to be measurable it is necessary and sufficient that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610103.png" /> there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610104.png" />, continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610105.png" />, such that the measure of the set
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For a function $f$ defined on the interval $[a,b]$ and almost-everywhere finite, to be [[Measurable function|measurable]] it is necessary and sufficient that for any $\epsilon>0$ there is a function $\phi$, continuous on $[a,b]$, such that the measure of the set
 
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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/l061010/l0610106.png" /></td> </tr></table>
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\{ x \in [a,b] : f(x) \ne \phi(x) \}
 
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$$
is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610107.png" />. It was proved by N.N. Luzin [[#References|[1]]]. In other words, an almost-everywhere finite function is measurable if and only if it becomes continuous if one neglects a set of arbitrary small measure.
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is less than $\epsilon$. It was proved by N.N. Luzin [[#References|[1]]]. In other words, an almost-everywhere finite function is measurable if and only if it becomes continuous if one neglects a set of arbitrary small measure.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,  "Sur les propriétés des fonctions mesurables"  ''C.R. Acad. Sci. Paris'' , '''154'''  (1912)  pp. 1688–1690  {{MR|}}  {{ZBL|43.0484.04}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)  {{MR|0640867}} {{MR|0354979}} {{MR|0148805}} {{MR|0067952}} {{MR|0039790}} {{ZBL|}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,  "Sur les propriétés des fonctions mesurables"  ''C.R. Acad. Sci. Paris'' , '''154'''  (1912)  pp. 1688–1690  {{MR|}}  {{ZBL|43.0484.04}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)  {{MR|0640867}} {{MR|0354979}} {{MR|0148805}} {{MR|0067952}} {{MR|0039790}} {{ZBL|}} </TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
In the West, Luzin's criterion is known as Luzin's theorem (in spite of an ambiguity — cf. [[Luzin theorem|Luzin theorem]]) and is generally stated a little bit differently, more like in [[Luzin-C-property|Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610108.png" />-property]] (but with a compact set instead of a perfect set). The tightness of the measure and the normality of the space makes all these formulations equivalent.
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In the West, Luzin's criterion is known as Luzin's theorem (in spite of an ambiguity — cf. [[Luzin theorem]]) and is generally stated a little bit differently, more like in [[Luzin-C-property|Luzin $C$-property]] (but with a compact set instead of a perfect set). The tightness of the measure and the normality of the space makes all these formulations equivalent.
  
The Luzin criterion remains true if the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610109.png" /> is replaced by any [[Completely-regular space|completely-regular space]] and the (restriction of the) Lebesgue measure by any tight bounded measure on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l06101010.png" />-field. In this general setting the Luzin property may be used in order to give an alternative definition of the notion of measurability (cf. [[#References|[a1]]]) or, in recent works, a more adequate definition of this notion when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l06101011.png" /> is no longer a real-valued function but, for example, a Banach-valued function.
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The Luzin criterion remains true if the interval $[a,b]$ is replaced by any [[completely-regular space]] and the (restriction of the) Lebesgue measure by any tight bounded measure on the Borel $\sigma$-field. In this general setting the Luzin property may be used in order to give an alternative definition of the notion of measurability (cf. [[#References|[a1]]]) or, in recent works, a more adequate definition of this notion when $f$ is no longer a real-valued function but, for example, a Banach-valued function.
  
The Luzin criterion is intimately related to the [[Egorov theorem|Egorov theorem]] and to the notion of measurability according to Carathéodory (cf. [[Carathéodory measure|Carathéodory measure]]).
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The Luzin criterion is intimately related to the [[Egorov theorem]] and to the notion of measurability according to Carathéodory (cf. [[Carathéodory measure]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)  {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1966)  pp. 98  {{MR|0210528}} {{ZBL|0142.01701}} </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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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)  {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1966)  pp. 98  {{MR|0210528}} {{ZBL|0142.01701}} </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>
 +
 
 +
{{TEX|done}}

Latest revision as of 19:43, 28 December 2017

for measurability of a function of a real variable

For a function $f$ defined on the interval $[a,b]$ and almost-everywhere finite, to be measurable it is necessary and sufficient that for any $\epsilon>0$ there is a function $\phi$, continuous on $[a,b]$, such that the measure of the set $$ \{ x \in [a,b] : f(x) \ne \phi(x) \} $$ is less than $\epsilon$. It was proved by N.N. Luzin [1]. In other words, an almost-everywhere finite function is measurable if and only if it becomes continuous if one neglects a set of arbitrary small measure.

References

[1] N.N. [N.N. Luzin] Lusin, "Sur les propriétés des fonctions mesurables" C.R. Acad. Sci. Paris , 154 (1912) pp. 1688–1690 Zbl 43.0484.04
[2] I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian) MR0640867 MR0354979 MR0148805 MR0067952 MR0039790


Comments

In the West, Luzin's criterion is known as Luzin's theorem (in spite of an ambiguity — cf. Luzin theorem) and is generally stated a little bit differently, more like in Luzin $C$-property (but with a compact set instead of a perfect set). The tightness of the measure and the normality of the space makes all these formulations equivalent.

The Luzin criterion remains true if the interval $[a,b]$ is replaced by any completely-regular space and the (restriction of the) Lebesgue measure by any tight bounded measure on the Borel $\sigma$-field. In this general setting the Luzin property may be used in order to give an alternative definition of the notion of measurability (cf. [a1]) or, in recent works, a more adequate definition of this notion when $f$ is no longer a real-valued function but, for example, a Banach-valued function.

The Luzin criterion is intimately related to the Egorov theorem and to the notion of measurability according to Carathéodory (cf. Carathéodory measure).

References

[a1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[a2] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[a3] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701
[a4] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
How to Cite This Entry:
Luzin criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_criterion&oldid=28240
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article