Difference between revisions of "Radon-Nikodým theorem"
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[[Category:Classical measure theory]] | [[Category:Classical measure theory]] | ||
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$\newcommand{\abs}[1]{\left|#1\right|}$ | $\newcommand{\abs}[1]{\left|#1\right|}$ | ||
| − | A classical theorem in measure theory first established by J. Radon and O.M. | + | A classical theorem in measure theory first established by J. Radon and O.M. Nikodým, which has the following statement. |
| − | Let $\mathcal{B}$ be a | + | '''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 | 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 | + | 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} | \begin{equation}\label{e:R-N} | ||
\nu (B) = \int_B f\, d\mu \qquad \forall B\in \mathcal{B}\, . | \nu (B) = \int_B f\, d\mu \qquad \forall B\in \mathcal{B}\, . | ||
\end{equation} | \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, | + | 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}}. |
| − | (see [[Signed measure]]). More precisely, let $\mu$ be a (nonnegative real-valued) measure on $\mathcal{B}$, $V$ be a finite-dimensional | + | The theorem can be generalized to signed measures, $\mathbb C$-valued measures and, more in general, measures taking values in a finite-dimensional space |
| − | vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive function such that $\nu (A) = 0$ whenever $\mu (A) =0$. | + | (see [[Signed measure]]). More precisely, let $\mu$ be a (nonnegative real-valued) $\sigma$-finite measure on $\mathcal{B}$, $V$ be a finite-dimensional |
| − | Then there is a function $f\in L^1 (\mu, V)$ such that \ref{e:R-N} | + | vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive [[Set function|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==== | ====References==== | ||
{| | {| | ||
|- | |- | ||
| − | |valign="top"|{{Ref| | + | |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|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}} | + | |valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}} |
|- | |- | ||
| − | |valign="top"|{{Ref| | + | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} |
|- | |- | ||
| − | |valign="top"|{{Ref| | + | |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|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}} | ||
Latest revision as of 12:09, 14 December 2012
2020 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=27252
Radon-Nikodým theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon-Nikod%C3%BDm_theorem&oldid=27252
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article