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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|28A15}}
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[[Category:Classical measure theory]]
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{{TEX|done}}
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$\newcommand{\abs}[1]{\left|#1\right|}$
  
<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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A classical theorem in measure theory first established by J. Radon and O.M. Nikodým, which has the following statement.  
  
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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'''Theorem 1'''
 +
Let $\mathcal{B}$ be a [[Algebra of sets|σ-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 $\mu (A) = 0$, and $\mu$ is $\sigma$-finite, 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}
  
====References====
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The function $f$ is uniquely determined up to sets of $\mu$-measure zero and the $\sigma$-finiteness assumption of $\mu$ is necessary. For a proof see for instance Section 31 of {{Cite|Ha}}.
<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>
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The theorem can be generalized to signed measures, $\mathbb C$-valued measures and, more in general, measures taking values in a finite-dimensional space
 
+
(see [[Signed measure]]). More precisely, let $\mu$ be a (nonnegative real-valued) $\sigma$-finite measure on $\mathcal{B}$, $V$ be a finite-dimensional
 
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vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive [[Set function|set function]] such that $\nu (A) = 0$ whenever $\mu (A) =0$.
 
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Then there is a function $f\in L^1 (\mu, V)$ such that \ref{e:R-N} holds. This statement can be generalized to some, but not all, Banach spaces. If the conclusion of Theorem 1 holds for measures $\nu$ taking values in a certain Banach space $B$, then $B$ is said to have the Radon-Nikodym property, see [[Vector measure]].
====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.
 
  
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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For a useful characterization of the density $f$ in the case of Radon measures in euclidean spaces see [[Differentiation of 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>
+
{|
 +
|-
 +
|valign="top"|{{Ref|AFP}}||  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}}
 +
|-
 +
|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of  probability measures" , Wiley (1968)  {{MR|0233396}}  {{ZBL|0172.21201}}
 +
|-
 +
|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}}
 +
|-
 +
|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,  '''1'''  , Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|HS}}|| E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
 +
|-
 +
|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}}
 +
|-
 +
|valign="top"|{{Ref|Ni}}|| O. M. Nikodym, "Sur une généralisation des intégrales de M. J. Radon".  ''Fund. Math.'' , '''15'''  (1930)  pp. 131–179
 +
|-
 +
|valign="top"|{{Ref|Ra}}|| J. Radon, "Ueber lineare Funktionaltransformationen und Funktionalgleichungen", ''Sitzungsber. Acad. Wiss. Wien'' , '''128'''  (1919)  pp. 1083–1121
 +
|-
 +
|}

Latest revision as of 12:09, 14 December 2012

2010 Mathematics Subject Classification: Primary: 28A15 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

A classical theorem in measure theory first established by J. Radon and O.M. Nikodým, which has the following statement.

Theorem 1 Let $\mathcal{B}$ be a σ-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 $\mu (A) = 0$, and $\mu$ is $\sigma$-finite, 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 and the $\sigma$-finiteness assumption of $\mu$ is necessary. For a proof see for instance Section 31 of [Ha]. The theorem can be generalized to signed measures, $\mathbb C$-valued measures and, more in general, measures taking values in a finite-dimensional space (see Signed measure). More precisely, let $\mu$ be a (nonnegative real-valued) $\sigma$-finite measure on $\mathcal{B}$, $V$ be a finite-dimensional vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive set 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} holds. This statement can be generalized to some, but not all, Banach spaces. If the conclusion of Theorem 1 holds for measures $\nu$ taking values in a certain Banach space $B$, then $B$ is said to have the Radon-Nikodym property, see Vector measure.

For a useful characterization of the density $f$ in the case of Radon measures in euclidean spaces see Differentiation of measures.

References

[AFP] 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
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[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 Zbl 0635.47001
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[HS] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[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=15021
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article