Radon-Nikodým theorem

generalized measure, real valued measure

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

A classical theorem in measure theory first established by J. Radon and O.M. Nikodym, which has the following statement.

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 $\mu (A) = 0$, 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.

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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