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''of a parallelepiped
 
''of a parallelepiped
  
<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/j/j054/j054350/j0543501.png" /></td> </tr></table>
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$$ \tag{* }
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\Delta  = \{ {x \in \mathbf R  ^ {n} } : {
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a _ {i} \leq  x _ {i} \leq  b _ {i} ,\
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a _ {i} < b _ {i} , i = 1 \dots n } \}
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$$
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543502.png" />''
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in $  \mathbf R  ^ {n} $''
  
The volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543503.png" /> of this parallelepiped. The following are defined for a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543504.png" />: the outer Jordan measure
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The volume $  m \Delta = \prod _ {i=} 1  ^ {n} ( b _ {i} - a _ {i} ) $
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of this parallelepiped. The following are defined for a bounded set $  E \subset  \mathbf R  ^ {n} $:  
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the outer Jordan measure
  
<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/j/j054/j054350/j0543505.png" /></td> </tr></table>
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$$
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m _ {e} E  = \inf  \sum _ { j= } 1 ^ { k }  m \Delta _ {j} ,\ \
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\cup _ { j= } 1 ^ { k }  \Delta _ {j} \supset E ,\ \
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k = 1 , 2 \dots
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$$
  
 
and the inner Jordan measure
 
and the inner Jordan measure
  
<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/j/j054/j054350/j0543506.png" /></td> </tr></table>
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$$
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m _ {i} E  = \sup  \sum _ { j= } 1 ^ { k }  m \Delta _ {j} ,\ \
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E \supset \Delta _ {j} ,
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$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543507.png" /> are pairwise disjoint (here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543508.png" /> are parallelepipeds of the form ). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543509.png" /> is said to be Jordan measurable (squarable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435010.png" />, cubable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435011.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435012.png" /> or, equivalently, if
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where the $  \Delta _ {j} $
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are pairwise disjoint (here the $  \Delta _ {j} $
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are parallelepipeds of the form ). A set $  E $
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is said to be Jordan measurable (squarable for $  n = 2 $,  
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cubable for $  n \geq  3 $)  
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if $  m _ {e} E = m _ {i} E $
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or, equivalently, 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/j/j054/j054350/j05435013.png" /></td> </tr></table>
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$$
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m _ {e} E + m _ {e} ( \Delta \setminus  E )  = m \Delta ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435014.png" />. In this case, the Jordan measure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435015.png" />. A bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435016.png" /> is Jordan measurable if and only if its boundary has Jordan measure zero (or, equivalently, if its boundary has [[Lebesgue measure|Lebesgue measure]] zero).
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where $  \Delta \supset E $.  
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In this case, the Jordan measure is $  m E = m _ {e} E = m _ {i} E $.  
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A bounded set $  E \subset  \mathbf R  ^ {n} $
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is Jordan measurable if and only if its boundary has Jordan measure zero (or, equivalently, if its boundary has [[Lebesgue measure|Lebesgue measure]] zero).
  
The concept of this measure was introduced by G. Peano [[#References|[1]]] and C. Jordan [[#References|[2]]]. The outer measure is the same for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435018.png" /> (the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435019.png" />, cf. [[Closure of a set|Closure of a set]]) and is equal to the [[Borel measure|Borel measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435020.png" />. The Jordan-measurable sets form a ring of sets on which the Jordan measure is a finitely-additive function. See also [[Squarability|Squarability]].
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The concept of this measure was introduced by G. Peano [[#References|[1]]] and C. Jordan [[#References|[2]]]. The outer measure is the same for $  E $
 +
and $  \overline{E}\; $(
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the closure of $  E $,  
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cf. [[Closure of a set|Closure of a set]]) and is equal to the [[Borel measure|Borel measure]] of $  \overline{E}\; $.  
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The Jordan-measurable sets form a ring of sets on which the Jordan measure is a finitely-additive function. See also [[Squarability|Squarability]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Peano,  "Applicazioni geometriche del calcolo infinitesimale" , Bocca  (1887)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Jordan,  "Remarques sur les intégrales définies"  ''J. Math. Pures Appl.'' , '''8'''  (1892)  pp. 69–99</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Peano,  "Applicazioni geometriche del calcolo infinitesimale" , Bocca  (1887)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Jordan,  "Remarques sur les intégrales définies"  ''J. Math. Pures Appl.'' , '''8'''  (1892)  pp. 69–99</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</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>
 
 
  
 
====Comments====
 
====Comments====
 
Jordan measure (also called Jordan content) is now mainly of historical interest, since it is simply the restriction of [[Lebesgue measure|Lebesgue measure]] to the ring of bounded Lebesgue-measurable sets having boundary of measure 0.
 
Jordan measure (also called Jordan content) is now mainly of historical interest, since it is simply the restriction of [[Lebesgue measure|Lebesgue measure]] to the ring of bounded Lebesgue-measurable sets having boundary of measure 0.

Latest revision as of 22:14, 5 June 2020


of a parallelepiped

$$ \tag{* } \Delta = \{ {x \in \mathbf R ^ {n} } : { a _ {i} \leq x _ {i} \leq b _ {i} ,\ a _ {i} < b _ {i} , i = 1 \dots n } \} $$

in $ \mathbf R ^ {n} $

The volume $ m \Delta = \prod _ {i=} 1 ^ {n} ( b _ {i} - a _ {i} ) $ of this parallelepiped. The following are defined for a bounded set $ E \subset \mathbf R ^ {n} $: the outer Jordan measure

$$ m _ {e} E = \inf \sum _ { j= } 1 ^ { k } m \Delta _ {j} ,\ \ \cup _ { j= } 1 ^ { k } \Delta _ {j} \supset E ,\ \ k = 1 , 2 \dots $$

and the inner Jordan measure

$$ m _ {i} E = \sup \sum _ { j= } 1 ^ { k } m \Delta _ {j} ,\ \ E \supset \Delta _ {j} , $$

where the $ \Delta _ {j} $ are pairwise disjoint (here the $ \Delta _ {j} $ are parallelepipeds of the form ). A set $ E $ is said to be Jordan measurable (squarable for $ n = 2 $, cubable for $ n \geq 3 $) if $ m _ {e} E = m _ {i} E $ or, equivalently, if

$$ m _ {e} E + m _ {e} ( \Delta \setminus E ) = m \Delta , $$

where $ \Delta \supset E $. In this case, the Jordan measure is $ m E = m _ {e} E = m _ {i} E $. A bounded set $ E \subset \mathbf R ^ {n} $ is Jordan measurable if and only if its boundary has Jordan measure zero (or, equivalently, if its boundary has Lebesgue measure zero).

The concept of this measure was introduced by G. Peano [1] and C. Jordan [2]. The outer measure is the same for $ E $ and $ \overline{E}\; $( the closure of $ E $, cf. Closure of a set) and is equal to the Borel measure of $ \overline{E}\; $. The Jordan-measurable sets form a ring of sets on which the Jordan measure is a finitely-additive function. See also Squarability.

References

[1] G. Peano, "Applicazioni geometriche del calcolo infinitesimale" , Bocca (1887)
[2] C. Jordan, "Remarques sur les intégrales définies" J. Math. Pures Appl. , 8 (1892) pp. 69–99
[3] S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)
[4] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)

Comments

Jordan measure (also called Jordan content) is now mainly of historical interest, since it is simply the restriction of Lebesgue measure to the ring of bounded Lebesgue-measurable sets having boundary of measure 0.

How to Cite This Entry:
Jordan measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_measure&oldid=47470
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article