# 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

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 **R**^{n}[edit]

As a particular case, a measure defined on the Euclidean space 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,

has the Dirac delta distribution as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure , nor is absolutely continuous with respect to : but ; if is any open set not containing 0, then but .

**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.

## See also[edit]

## References[edit]

- 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.*