Singular measures
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 $\nu$.
Comments
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.
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 |
[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 |
[Bi] | P. Billingsley, "Convergence of probability measures", Wiley (1968) MR0233396 Zbl 0172.21201 |
[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 |
Mutually-singular measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mutually-singular_measures&oldid=27496