Singular measure explained

In mathematics, two positive (or signed or complex) measures

and defined on a measurable space are called singular if there exist two disjoint measurable sets whose union is such that is zero on all measurable subsets of while is zero on all measurable subsets of 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 Rⁿ

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.

as its distributional derivative. This is a measure on the real line, a "point mass" at However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure nor is absolutely continuous with respect to but if is any non-empty 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.

Example. A singular continuous measure on

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

References

• Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. .
• J Taylor, An Introduction to Measure and Probability, Springer, 1996. .