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| In [[mathematics]], two positive (or [[signed measure|signed]] or [[complex measure|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 [[set union|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 <math>\mu \perp \nu.</math>
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| 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.
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| ==Examples on '''R'''<sup>''n''</sup>==
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| As a particular case, a measure defined on the [[Euclidean space]] '''R'''<sup>''n''</sup> 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.
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| '''Example.''' A [[discrete measure]].
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| The [[Heaviside step function]] on the [[real line]],
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| : <math>H(x) \ \stackrel{\mathrm{def}}{=} \begin{cases} 0, & x < 0; \\ 1, & x \geq 0; \end{cases}</math>
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| has the [[Dirac delta function|Dirac delta distribution]] <math>\delta_0</math> as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the [[Dirac measure]] <math>\delta_0</math> is not absolutely continuous with respect to Lebesgue measure <math>\lambda</math>, nor is <math>\lambda</math> absolutely continuous with respect to <math>\delta_0</math>: <math>\lambda ( \{ 0 \} ) = 0</math> but <math>\delta_0 ( \{ 0 \} ) = 1</math>; if <math>U</math> is any [[open set]] not containing 0, then <math>\lambda (U) > 0</math> but <math>\delta_0 (U) = 0</math>.
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| '''Example.''' A singular continuous measure.
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| 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.
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| ==See also== | |
| * [[Lebesgue's decomposition theorem]]
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| * [[Absolutely continuous]]
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| * [[Singular distribution]]
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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.
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| * J Taylor, ''An Introduction to Measure and Probability'', Springer, 1996. ISBN 0-387-94830-9.
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| {{PlanetMath attribution|id=4002|title=singular measure}}
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| [[Category:Integral calculus]]
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| [[Category:Measures (measure theory)]]
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