Singular measures

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2020 Mathematics Subject Classification: Primary: 28A15 [MSN][ZBL]

If $\mu$ and $\nu$ are two $\sigma$-finite measures on the same $\sigma$-algebra $\mathcal{B}$ of subsets of $X$, then $\mu$ and $\nu$ are said to be singular (or also mutually singular, or orthogonal) if there are two sets $A,B\in\mathcal{B}$ such that $A\cap B=\emptyset$, $A\cup B = X$ and $\mu (B)=\nu (A) = 0$. The concept can be extended to signed measures or vector-valued measures: in this case it is required that $\mu (B\cap E) = \nu (A\cap E) = 0$ for every $E\in\mathcal{B}$ (cp. with Section 30 of [Ha]). The singularity of the two measures $\mu$ and $\nu$ is usually denoted by $\mu\perp\nu$.

For general, i.e. non $\sigma$-finite (nonnegative) measures, the concept can be generalized in the following way: $\mu$ and $\nu$ are singular if the only (nonnegative) measure $\alpha$ on $\mathcal{B}$ with the property \[ \alpha (A)\leq \min \{\mu (A), \nu (A)\} \qquad \forall A\in\mathcal{B} \] is the trivial measure which assigns the value $0$ to every element of $\mathcal{B}$. By the Radon-Nikodym decomposition this concept coincides with the previous one when we assume the $\sigma$-finiteness of $\mu$ and $\nu$.


When $X$ is the standard euclidean space and $\mathcal{B}$ the Borel $\sigma$-algebra, the name singular measures is often used for those $\sigma$-finite measures $\mu$ which are orthogonal to the Lebesgue measure.


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Singular measures. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article