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 ${\displaystyle \mu \perp \nu .}$

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

As a particular case, a measure defined on the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ 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,

${\displaystyle H(x)\ {\stackrel {\mathrm {def} }{=}}{\begin{cases}0,&x<0;\\1,&x\geq 0;\end{cases}}}$

has the Dirac delta distribution ${\displaystyle \delta _{0}}$ as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure ${\displaystyle \delta _{0}}$ is not absolutely continuous with respect to Lebesgue measure ${\displaystyle \lambda }$, nor is ${\displaystyle \lambda }$ absolutely continuous with respect to ${\displaystyle \delta _{0}}$: ${\displaystyle \lambda (\{0\})=0}$ but ${\displaystyle \delta _{0}(\{0\})=1}$; if ${\displaystyle U}$ is any open set not containing 0, then ${\displaystyle \lambda (U)>0}$ but ${\displaystyle \delta _{0}(U)=0}$.

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 R².

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

See also

• Lebesgue's decomposition theorem
• Absolutely continuous
• Singular distribution

References

• 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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