Singular measure

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 Rⁿ is called singular, if it is singular in 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.

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Singular measure

Examples on Rn

See also

References

Examples on Rⁿ