View source for Radon-Nikodým theorem

{{MSC|28A15}}
[[Category:Classical measure theory]]
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A classical theorem in measure theory first established by J. Radon and O.M. Nikod\'ym, which has the following statement. 

'''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
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 {{Cite|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 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. 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====
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|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}}
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|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}}
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|valign="top"|{{Ref|DS}}||   N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,  '''1'''  , Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|valign="top"|{{Ref|HS}}||  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
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|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}}
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|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
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