In mathematics, two positive (or signed or complex) measures
defined on a measurable space
are called singular if there exist two disjoint measurable sets
A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure.
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.
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
is not absolutely continuous with respect to Lebesgue measure
absolutely continuous with respect to
is any non-empty open set not containing 0, then
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.
A singular continuous measure on
The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
This article incorporates material from singular measure on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.