In mathematics, two positive (or signed or complex defined on a measurable space are called singular whose union

A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure.

Examples on Rn

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 However, the Dirac measure is any non-empty open set

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.

The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.

See also

References

• Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. .

• J Taylor, An Introduction to Measure and Probability, Springer, 1996. .