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singular measure

In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on all measurable subsets of B while \nu 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 '''R'''<sup>''n''</sup>

As a particular case, a measure defined on the Euclidean space \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,

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

has the Dirac delta distribution \delta_0 as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure \delta_0 is not absolutely continuous with respect to Lebesgue measure \lambda, nor is \lambda absolutely continuous with respect to \delta_0: \lambda(\{0\}) = 0 but \delta_0(\{0\}) = 1; if U is any non-empty open set not containing 0, then \lambda(U) > 0 but \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 \R^2.

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

See also

  • {{annotated link|Absolute continuity (measure theory)}}
  • {{annotated link|Lebesgue's decomposition theorem}}
  • {{annotated link|Singular distribution}}

References

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{{reflist|group=note}}

  • 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}}.

{{Measure theory}}

{{PlanetMath attribution|id=4002|title=singular measure}}

Category:Integral calculus

Category:Measures (measure theory)