Difference between revisions of "Radon-Nikodým theorem"
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| − | + | [[Category:Classical measure theory]] | |
| − | + | {{TEX|done}} | |
| − | + | ''generalized measure'', ''real valued measure'' | |
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| − | + | $\newcommand{\abs}[1]{\left|#1\right|}$ | |
| + | 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==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |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|Bi}}|| P. Billingsley, "Convergence of probability measures" , Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|He}}|| E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) | ||
| + | |- | ||
| + | |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 | ||
| + | |- | ||
| + | |} | ||
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
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