In mathematics, two positive (or signed or complex) measures μ and ν defined on a measurable space (Ω, Σ) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. This is denoted by \( \mu \perp \nu \) .
A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples.
Examples on Rn
As a particular case, a measure defined on the Euclidean space \( \mathbb {R} ^{n} \) is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure.
Example. A discrete measure.
The Heaviside step function on the real line,
\( H(x)\ {\stackrel {{\mathrm {def}}}{=}}{\begin{cases}0,&x<0;\\1,&x\geq 0;\end{cases}} \)
has the Dirac delta distribution \( \delta _{0} \) as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure \( \delta _{0} \) is not absolutely continuous with respect to Lebesgue measure \( \lambda \) , nor is \( \lambda \) absolutely continuous with respect to \( \delta _{0} \) : \( \lambda (\{0\})=0 \) but \( \delta _{0}(\{0\})=1 \); if U is any open set not containing 0, then \( \lambda (U)>0 \) but \( \delta _{0}(U)=0. \)
Example. A singular continuous measure.
The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous.
Example. A singular continuous measure on R2.
The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
Lebesgue's decomposition theorem
Absolutely continuous
Singular distribution
References
Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. ISBN 1-58488-347-2.
J Taylor, An Introduction to Measure and Probability, Springer, 1996. ISBN 0-387-94830-9.
This article incorporates material from singular measure on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Undergraduate Texts in Mathematics
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