Radon-Nikodým theorem

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.


[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:
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article