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 $\mu \perp \nu.$

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 Rⁿ is called singular, if it is singular in 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,

$H(x) \ \stackrel{\mathrm{def}}{=} \ \left\{ \begin{matrix} 0, & x < 0; \\ 1, & x \geq 0; \end{matrix} \right.$

has the Dirac delta distribution δ₀ as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure δ₀ is not absolutely continuous with respect to Lebesgue measure λ, nor is λ absolutely continuous with respect to δ₀: λ({0}) = 0 but δ₀({0}) = 1; if U is any open set not containing 0, then λ(U) > 0 but δ₀(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.

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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