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''in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578701.png" />''
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A countably-additive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578702.png" /> which is an extension of the volume as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578703.png" />-dimensional intervals to a wider class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578704.png" /> of sets, namely the Lebesgue-measurable sets. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578705.png" /> contains the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578706.png" /> of Borel sets (cf. [[Borel set|Borel set]]) and consists of all sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578707.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l0578709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787010.png" />. One has for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787011.png" />,
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{{TEX|done}}
  
<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/l057/l057870/l05787012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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''in  $  \mathbf R  ^ {n} $''
  
where the infimum is taken over all possible countable families of intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787014.png" />. Formula (*) makes sense for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787015.png" /> and defines a set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787016.png" /> (which coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787018.png" />), called the outer Lebesgue measure. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787019.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787020.png" /> if and only if
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A countably-additive measure  $  \lambda $
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which is an extension of the volume as a function of $  n $-dimensional intervals to a wider class  $  {\mathcal A} $
 +
of sets, namely the Lebesgue-measurable sets. The class  $  {\mathcal A} $
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contains the class  $  {\mathcal B} $
 +
of Borel sets (cf. [[Borel set|Borel set]]) and consists of all sets of the form  $  A \cup B $
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where  $  B \subset  B _ {1} $,  
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A , B _ {1} \in {\mathcal B} $
 +
and  $  \lambda ( B _ {1} ) = 0 $.  
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One has for any  $  A \in {\mathcal A} $,
  
<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/l057/l057870/l05787021.png" /></td> </tr></table>
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$$ \tag{* }
 +
\lambda ( A)  = \inf  \sum _ { j } \lambda ( I _ {j} ) ,
 +
$$
  
for every bounded interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787022.png" />; for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787023.png" />,
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where the infimum is taken over all possible countable families of intervals  $  \{ I _ {j} \} $
 +
such that  $  A \subset  \cup I _ {j} $.
 +
Formula (*) makes sense for every $  A \subset  \mathbf R  ^ {n} $
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and defines a set function  $  \lambda  ^ {*} $(
 +
which coincides with  $  \lambda $
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on  $  {\mathcal A} $),
 +
called the outer Lebesgue measure. A set  $  A $
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belongs to  $  {\mathcal A} $
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if and only if
  
<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/l057/l057870/l05787024.png" /></td> </tr></table>
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$$
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\lambda ( I)  = \lambda  ^ {*} ( A \cap I ) + \lambda  ^ {*} ( I \setminus  A )
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$$
  
and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787025.png" />,
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for every bounded interval  $  I $;
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for all $  A \subset  \mathbf R  ^ {n} $,
  
<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/l057/l057870/l05787026.png" /></td> </tr></table>
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$$
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\lambda  ^ {*} ( A)  = \inf
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\{ {\lambda ( U ) } : {A \subset  U , U  \textrm{ is open} } \}
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,
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$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787027.png" />, then the last equality is sufficient for the membership <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787028.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787029.png" /> is an orthogonal operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787032.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787033.png" />. The Lebesgue measure was introduced by H. Lebesgue [[#References|[1]]].
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and for all  $  A \in {\mathcal A} $,
  
====References====
+
$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Intégrale, longeur, aire" , Univ. Paris  (1902(Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Saks,  "Theory of the integral" , Hafner (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand (1950)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian)</TD></TR></table>
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\lambda ( A)  =  \lambda ^ {*} ( A= \
 +
\sup \{ {\lambda ( F ) } : {A \supset F , F  \textrm{ is compact}
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  } \}
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;
 +
$$
  
 +
if  $  \lambda  ^ {*} ( A) < \infty $,
 +
then the last equality is sufficient for the membership  $  A \in {\mathcal A} $;
 +
if  $  O $
 +
is an orthogonal operator in  $  \mathbf R  ^ {n} $
 +
and  $  a \in \mathbf R  ^ {n} $,
 +
then  $  \lambda ( OA + a ) = \lambda ( A) $
 +
for any  $  A \in {\mathcal A} $.
 +
The Lebesgue measure was introduced by H. Lebesgue [[#References|[1]]].
  
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Intégrale, longeur, aire" , Univ. Paris  (1902)  (Thesis)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
The Lebesgue measure is a very particular example of a [[Haar measure|Haar measure]], of a product measure (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057870/l05787034.png" />) and of a [[Hausdorff measure|Hausdorff measure]]. Actually it is historically the first example of such measures.
+
The Lebesgue measure is a very particular example of a [[Haar measure|Haar measure]], of a product measure (when $  n > 1 $)  
 +
and of a [[Hausdorff measure|Hausdorff measure]]. Actually it is historically the first example of such measures.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</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 16:34, 5 February 2022


in $ \mathbf R ^ {n} $

A countably-additive measure $ \lambda $ which is an extension of the volume as a function of $ n $-dimensional intervals to a wider class $ {\mathcal A} $ of sets, namely the Lebesgue-measurable sets. The class $ {\mathcal A} $ contains the class $ {\mathcal B} $ of Borel sets (cf. Borel set) and consists of all sets of the form $ A \cup B $ where $ B \subset B _ {1} $, $ A , B _ {1} \in {\mathcal B} $ and $ \lambda ( B _ {1} ) = 0 $. One has for any $ A \in {\mathcal A} $,

$$ \tag{* } \lambda ( A) = \inf \sum _ { j } \lambda ( I _ {j} ) , $$

where the infimum is taken over all possible countable families of intervals $ \{ I _ {j} \} $ such that $ A \subset \cup I _ {j} $. Formula (*) makes sense for every $ A \subset \mathbf R ^ {n} $ and defines a set function $ \lambda ^ {*} $( which coincides with $ \lambda $ on $ {\mathcal A} $), called the outer Lebesgue measure. A set $ A $ belongs to $ {\mathcal A} $ if and only if

$$ \lambda ( I) = \lambda ^ {*} ( A \cap I ) + \lambda ^ {*} ( I \setminus A ) $$

for every bounded interval $ I $; for all $ A \subset \mathbf R ^ {n} $,

$$ \lambda ^ {*} ( A) = \inf \{ {\lambda ( U ) } : {A \subset U , U \textrm{ is open} } \} , $$

and for all $ A \in {\mathcal A} $,

$$ \lambda ( A) = \lambda ^ {*} ( A) = \ \sup \{ {\lambda ( F ) } : {A \supset F , F \textrm{ is compact} } \} ; $$

if $ \lambda ^ {*} ( A) < \infty $, then the last equality is sufficient for the membership $ A \in {\mathcal A} $; if $ O $ is an orthogonal operator in $ \mathbf R ^ {n} $ and $ a \in \mathbf R ^ {n} $, then $ \lambda ( OA + a ) = \lambda ( A) $ for any $ A \in {\mathcal A} $. The Lebesgue measure was introduced by H. Lebesgue [1].

References

[1] H. Lebesgue, "Intégrale, longeur, aire" , Univ. Paris (1902) (Thesis)
[2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[3] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[4] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801

Comments

The Lebesgue measure is a very particular example of a Haar measure, of a product measure (when $ n > 1 $) and of a Hausdorff measure. Actually it is historically the first example of such measures.

References

[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
How to Cite This Entry:
Lebesgue measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_measure&oldid=15438
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article