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

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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   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.

See also

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References

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  • 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.

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