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Difference between revisions of "Carathéodory measure"

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The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203301.png" /> induced by the outer Carathéodory measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203302.png" />, the latter being an [[Outer measure|outer measure]] defined on the class of all subsets of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203303.png" /> (with a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203304.png" />) such that
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{{MSC|28A}}
  
<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/c/c020/c020330/c0203305.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203306.png" />. It was introduced by C. Carathéodory [[#References|[1]]]. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203307.png" /> belongs to the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203308.png" />, i.e. is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c0203309.png" />-measurable, if and only if
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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/c/c020/c020330/c02033010.png" /></td> </tr></table>
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The term might refer to different objects in classical measure theory.
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====Caratheodory measures and outer measures in metric spaces====
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Consider an [[Outer measure|outer measure]] $\mu$ defined on the class $\mathcal{P} (X)$ of subsets of a metric space $(X,d)$. $\mu$ is a ''Caratheodory outer measure'', more often called ''metric outer measure'' (cp. with Section 11 of {{Cite|Ha}}), if
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\begin{equation}\label{e:additive}
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\mu (A\cup B) = \mu (A) + \mu (B)
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\end{equation}
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for every pair of sets $A, B\subset X$ which have positive distance (i.e. such that $\inf \{d(x,y): x\in A, y\in B\} > 0$). A theorem due to Caratheodory shows then that the Borel sets are $\mu$-measurable (see [[Outer measure#Caratheodory criterion]], also for the notion of $\mu$-measurability). The restriction of $\mu$ to the [[Algebra of sets|$\sigma$-algebra]] of $\mu$-measurable sets is called, by some authors, the Caratheodory measure induced by the metric outer measure $\mu$.
  
for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033011.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033012.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033014.png" />-measurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033015.png" />. The domain of definition of a Carathéodory measure contains all Borel sets. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033016.png" /> is an outer measure on the class of all subsets of a metric space such that every open set is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033017.png" />-measurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020330/c02033018.png" /> is an outer Carathéodory measure.
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The converse is also true: if $\mu$ is an outer measure on a metric space $(X,d)$ for which the open set are $\mu$-measurable, then $\mu$ is a metric outer measure (see for instance Remark (8c) of Section 11 in {{Cite|Ha}}).
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  "Ueber das lineare Mass von Punktmengen, eine Verallgemeinerung des Längenbegriffs"  ''Nachr. Gesell. Wiss. Göttingen''  (1914) pp. 404–426</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></table>
 
  
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====Caratheodory outer measures with respect to a class of functions====
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More generally, given a set $X$ and a class $\Gamma$ of real functions on $X$, some authors (see for instance Section 7 of Chapter 12 in {{Cite|Ro}}) call ''Caratheodory outer measures with respect to $\Gamma$'' those outer measures $\mu$ on $\mathcal{P} (X)$ with the property that \eqref{e:additive} holds when $A$ and $B$ are ''separated by $\Gamma$'', i.e. when there is a function $\varphi\in \Gamma$ with $\inf_A\; \varphi > \sup_B \varphi$ or $\inf_B\;\varphi > \sup_B\; \varphi$.
  
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If $(X,d)$ is a metric space and we chose as $\Gamma$ the set of functions of type $x\mapsto {\rm dist}\, (x, E)$ with $E\subset X$, then a ''Caratheodory outer measure with respect to $\Gamma$'' corresponds to a Caratheodory outer measure in the sense of the previous section.
  
====Comments====
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====Caratheodory (outer) measures in the Euclidean space====
An outer Carathéodory measure is also frequently called a metric outer measure, cf. [[#References|[a1]]].
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Some authors use the term ''Caratheodory (outer) measures'' for a special class of outer measures defined on the subsets of the euclidean space $\mathbb R^n$ and constructed in a fashion similar to the usual [[Hausdorff measure|Hausdorff (outer) measures]]. Cp. for instance with  Sections  2.1.3-2.1.4-2.1.5 of {{Cite|KP}} and Sections 2.10.2-2.10.3-2.10.4 of {{Cite|Fe}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. HewittK.R. Stromberg,  "Real and abstract analysis" , Springer (1965)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ca}}|| C. Carathéodory,  "Über das lineare Mass von Punktmengen, eine Verallgemeinerung des Längenbegriffs"  ''Nachr. Gesell. Wiss. Göttingen''  (1914)  pp. 404–426.
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|-
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|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of    functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,    1992. {{MR|1158660}} {{ZBL|0804.2800}}
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|-
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|valign="top"|{{Ref|Fe}}||    H. Federer, "Geometric measure  theory". Volume 153 of Die  Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New  York Inc., New   York, 1969.  {{MR|0257325}} {{ZBL|0874.49001}}
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|-
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|KP}}|| S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
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|-
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|valign="top"|{{Ref|Ma}}||      P. Mattila, "Geometry of sets  and measures in Euclidean spaces.  Fractals and rectifiability".    Cambridge Studies in Advanced  Mathematics, 44. Cambridge University      Press, Cambridge,  1995.  {{MR|1333890}} {{ZBL|0911.28005}}
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|-
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|valign="top"|{{Ref|Mu}}|| M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
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|-
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan (1969). {{MR|0151555}} {{ZBL|0197.03501}}
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|-
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|}

Latest revision as of 09:27, 7 December 2012

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

The term might refer to different objects in classical measure theory.

Caratheodory measures and outer measures in metric spaces

Consider an outer measure $\mu$ defined on the class $\mathcal{P} (X)$ of subsets of a metric space $(X,d)$. $\mu$ is a Caratheodory outer measure, more often called metric outer measure (cp. with Section 11 of [Ha]), if \begin{equation}\label{e:additive} \mu (A\cup B) = \mu (A) + \mu (B) \end{equation} for every pair of sets $A, B\subset X$ which have positive distance (i.e. such that $\inf \{d(x,y): x\in A, y\in B\} > 0$). A theorem due to Caratheodory shows then that the Borel sets are $\mu$-measurable (see Outer measure#Caratheodory criterion, also for the notion of $\mu$-measurability). The restriction of $\mu$ to the $\sigma$-algebra of $\mu$-measurable sets is called, by some authors, the Caratheodory measure induced by the metric outer measure $\mu$.

The converse is also true: if $\mu$ is an outer measure on a metric space $(X,d)$ for which the open set are $\mu$-measurable, then $\mu$ is a metric outer measure (see for instance Remark (8c) of Section 11 in [Ha]).

Caratheodory outer measures with respect to a class of functions

More generally, given a set $X$ and a class $\Gamma$ of real functions on $X$, some authors (see for instance Section 7 of Chapter 12 in [Ro]) call Caratheodory outer measures with respect to $\Gamma$ those outer measures $\mu$ on $\mathcal{P} (X)$ with the property that \eqref{e:additive} holds when $A$ and $B$ are separated by $\Gamma$, i.e. when there is a function $\varphi\in \Gamma$ with $\inf_A\; \varphi > \sup_B \varphi$ or $\inf_B\;\varphi > \sup_B\; \varphi$.

If $(X,d)$ is a metric space and we chose as $\Gamma$ the set of functions of type $x\mapsto {\rm dist}\, (x, E)$ with $E\subset X$, then a Caratheodory outer measure with respect to $\Gamma$ corresponds to a Caratheodory outer measure in the sense of the previous section.

Caratheodory (outer) measures in the Euclidean space

Some authors use the term Caratheodory (outer) measures for a special class of outer measures defined on the subsets of the euclidean space $\mathbb R^n$ and constructed in a fashion similar to the usual Hausdorff (outer) measures. Cp. for instance with Sections 2.1.3-2.1.4-2.1.5 of [KP] and Sections 2.10.2-2.10.3-2.10.4 of [Fe].

References

[Ca] C. Carathéodory, "Über das lineare Mass von Punktmengen, eine Verallgemeinerung des Längenbegriffs" Nachr. Gesell. Wiss. Göttingen (1914) pp. 404–426.
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[KP] S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
[Ma] P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Mu] M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501
How to Cite This Entry:
Carathéodory measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_measure&oldid=23222
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article