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A [[Charge|charge]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771801.png" /> that is absolutely continuous with respect to some measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771802.png" /> has a density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771803.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771804.png" /> that is summable with respect to this measure. It was established by J. Radon [[#References|[1]]] and O.M. Nikodým [[#References|[2]]]. More precisely, on a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771805.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771806.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771807.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771808.png" />, suppose one is given a charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r0771809.png" />, i.e. a countably-additive real or complex function given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718010.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718011.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718012.png" />, and, moreover, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718013.png" /> be absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718014.png" />. Then there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718016.png" />, summable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718017.png" />, such that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718018.png" />,
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{{MSC|28A33}}
  
<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/r/r077/r077180/r07718019.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718020.png" /> is unique (except for modifications on a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718021.png" />-measure zero), and is called the density of the charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718022.png" /> with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718023.png" />. There are (see [[#References|[4]]]) generalizations of the theorem to the case when the charge takes values in some vector space.
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{{TEX|done}}
  
====References====
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''generalized measure'', ''real valued measure''
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Radon,  "Ueber lineare Funktionaltransformationen und Funktionalgleichungen"  ''Sitzungsber. Acad. Wiss. Wien'' , '''128'''  (1919)  pp. 1083–1121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.M. Nikodým,  "Sur une généralisation des intégrales de M. J. Radon"  ''Fund. Math.'' , '''15'''  (1930)  pp. 131–179</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Diestel,  J.J. Uhl jr.,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
The notion of  "charge"  is not well established in the West; one usually says  "signed measure"  (cf. [[Charge|Charge]]). The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718024.png" /> is also well defined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718025.png" /> is the sum of a series of (non-negative) measures; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718026.png" /> and the integral may take the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718027.png" />.
 
  
The theorem is false if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077180/r07718028.png" /> fails to satisfy same finiteness condition; see [[#References|[a1]]], §19, for a thorough discussion and illuminating examples.
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$\newcommand{\abs}[1]{\left|#1\right|}$
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A classical theorem in measure theory first established bz J. Radon and O.M. Nikodym, which has the following statement.  
  
For the generalizations of the theorem to vector measures (and relations to the geometry of Banach spaces) see [[Vector measure|Vector measure]].
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Let $\mathcal{B}$ be a $\sigma$-algebra of subsets
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of a set $X$ and let $\mu$ and $\nu$ be two measures on $\mathcal{B}$. If $\nu$ is absolutely continuous
 +
with respect to $\mu$, i.e. $\nu (A)=0$ whenever $\mu (A) = 0$, then there is a $\mathcal{B}$-measurable nonnegative function $f$ such that
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\begin{equation}\label{e:R-N}
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\nu (B) = \int_B f\, d\mu \qquad \forall B\in \mathcal{B}\, .
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\end{equation}
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The function $f$ is uniquely determined up to sets of $\mu$-measure zero.
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The theorem can be generalized to signed measures, $\mathbb C$-valued measures and, more in general, vector-valued measures
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(see [[Signed measure]]). More precisely, let $\mu$ be a (nonnegative real-valued) measure on $\mathcal{B}$, $V$ be a finite-dimensional
 +
vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive function such that $\nu (A) = 0$ whenever $\mu (A) =0$.
 +
Then there is a function $f\in L^1 (\mu, V)$ such that \ref{e:R-N} hold.
  
 
====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>
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{|
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|-
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|valign="top"|{{Ref|AmFuPa}}||  L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations  and  free discontinuity problems". Oxford Mathematical Monographs. The  Clarendon Press, Oxford University Press, New York, 2000.  {{MR|1857292}}{{ZBL|0957.49001}}
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|-
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|valign="top"|{{Ref|Bo}}||  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}}
 +
|-
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|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,  '''1'''  , Interscience (1958) {{MR|0117523}}
 +
|-
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|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
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|-
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|valign="top"|{{Ref|He}}|| E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)
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|-
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|valign="top"|{{Ref|Ma}}||  P. Mattila, "Geometry of sets and measures in euclidean spaces".  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|Ni}}|| O. M. Nikodym, "Sur une généralisation des intégrales de M. J. Radon".  ''Fund. Math.'' , '''15'''  (1930)  pp. 131–179
 +
|-
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|valign="top"|{{Ref|Ra}}|| J. Radon, "Ueber lineare Funktionaltransformationen und Funktionalgleichungen",
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''Sitzungsber. Acad. Wiss. Wien'' , '''128'''  (1919)  pp. 1083–1121
 +
|-
 +
|}

Revision as of 22:35, 27 July 2012

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

generalized measure, real valued measure

$\newcommand{\abs}[1]{\left|#1\right|}$ A classical theorem in measure theory first established bz J. Radon and O.M. Nikodym, which has the following statement.

Let $\mathcal{B}$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu$ and $\nu$ be two measures on $\mathcal{B}$. If $\nu$ is absolutely continuous with respect to $\mu$, i.e. $\nu (A)=0$ whenever $\mu (A) = 0$, then there is a $\mathcal{B}$-measurable nonnegative function $f$ such that \begin{equation}\label{e:R-N} \nu (B) = \int_B f\, d\mu \qquad \forall B\in \mathcal{B}\, . \end{equation} The function $f$ is uniquely determined up to sets of $\mu$-measure zero. The theorem can be generalized to signed measures, $\mathbb C$-valued measures and, more in general, vector-valued measures (see Signed measure). More precisely, let $\mu$ be a (nonnegative real-valued) measure on $\mathcal{B}$, $V$ be a finite-dimensional vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive function such that $\nu (A) = 0$ whenever $\mu (A) =0$. Then there is a function $f\in L^1 (\mu, V)$ such that \ref{e:R-N} hold.

References

[AmFuPa] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Bo] 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
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[He] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ni] O. M. Nikodym, "Sur une généralisation des intégrales de M. J. Radon". Fund. Math. , 15 (1930) pp. 131–179
[Ra] J. Radon, "Ueber lineare Funktionaltransformationen und Funktionalgleichungen",

Sitzungsber. Acad. Wiss. Wien , 128 (1919) pp. 1083–1121

How to Cite This Entry:
Radon-Nikodým theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon-Nikod%C3%BDm_theorem&oldid=27224
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article