Difference between revisions of "Radon-Nikodým theorem"
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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 | (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 $\mu (A) =0$. | 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. | + | Then there is a function $f\in L^1 (\mu, V)$ such that \ref{e:R-N} hold. See also [[Vector measure]] for more general statements. |
====References==== | ====References==== | ||
Revision as of 22:40, 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. See also Vector measure for more general statements.
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 |
Radon-Nikodým theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon-Nikod%C3%BDm_theorem&oldid=27224