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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 $$\label{e:R-N} \nu (B) = \int_B f\, d\mu \qquad \forall B\in \mathcal{B}\, .$$

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
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